Application of Capacities to Space-Time Fractional Dissipative Equations II: Carleson Measure Characterization for 


Lq(ℝ+n+1,μ)


−Extension

Pengtao Li; Zhichun Zhai

doi:10.1515/anona-2021-0232

Open Access Published by De Gruyter February 14, 2022

Application of Capacities to Space-Time Fractional Dissipative Equations II: Carleson Measure Characterization for Lq(ℝ+n+1,μ) −Extension

Pengtao Li and Zhichun Zhai

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2021-0232

Abstract

This paper provides the Carleson characterization of the extension of fractional Sobolev spaces and Lebesgue spaces to Lq(ℝ+n+1,μ) via space-time fractional equations. For the extension of fractional Sobolev spaces, preliminary results including estimates, involving the fractional capacity, measures, the non-tangential maximal function, and an estimate of the Riesz integral of the space-time fractional heat kernel, are provided. For the extension of Lebesgue spaces, a new L^p–capacity associated to the spatial-time fractional equations is introduced. Then, some basic properties of the L^p–capacity, including its dual form, the L^p–capacity of fractional parabolic balls, strong and weak type inequalities, are established.

Keywords: Space-time fractional equations; capacities; extension; fractional Laplacian; Lebesgue spaces; Sobolev spaces

MSC 2010: Primary: 31C15; Secondary: 26A33; 46E30

1 Introduction

In this paper, we study the Carleson characterization of Lq(ℝ+n+1,μ) -extensions of Sobolev spaces and Lebesgue spaces via the following space-time fractional equation:

(1.1) {∂tβu(x,t)=−ν(−Δ)α/2u(x,t)(x,t)∈ℝ+n+1;u(x,0)=φ(x),x∈ℝn.

with β ∈ (0, 1) and α > 0. Here the symbol ∂tβ denotes the Caputo fractional derivative defined as

∂tβu(x,t)=1Γ(1−β)∫0t∂rur(x)dr(t−r)β.

Carleson measures were introduced by L. Carleson [5] to characterize the interpolating sequences in the algebra H^∞ of bounded analytic functions in the open unit disc and to give a solution to the corona problem. In the viewpoint of geometry, a Carleson measure on a domain Ω can be seen as a measure that does not vanish at the boundary ∂Ω when compared to the surface measure on the boundary of Ω. In the field of harmonic analysis and the potential theory, Carleson measures are widely applied to the extension and the trace problem of function spaces. Let D denote the unit disc in the complex plane ℂ and µ be a Borel measure on D. For 1 ≤ p < ∞, let H^p(∂D) denote the Hardy space on the boundary of D and let L^p(D, µ) denote the Lebesgue space on D with respect to the measure µ. It is well known that the Poisson operator is bounded from H^p(∂D) to L^p(D, µ) if and only if µ is a Carleson measure, i.e., there exists a constant C > 0 such that, for every point x ∈ ∂D and every radius r > 0,

(1.2) μ(Ω∩B(x,r))/|∂Ω∩B(x,r)|≤C,

where B(x, r) denotes the ball centered at x with radius r. In the setting of the upper half space ℝ +n+1 , an analogue conclusion holds, see for instance [12, Theorem 7.37].

In 1960s, the capacities related with function spaces were introduced to characterize the embedding or trace inequalities. Let C0∞(ℝn) stand for all infinitely smooth functions with compact support in ℝⁿ. For a nonnegative Borel measure µ on ℝⁿ, to investigate the spectral problems for Schrödinger operators, Maz’ya first proved (cf. [14, 15]) that if 1 < p ≤ q and pl < n, then there exists a constant C such that

(1.3) ∥u∥Lq(ℝn,μ)≤C∥u∥h pl(ℝn), ∀ u∈h pl(ℝn)

holds if and only if

(1.4) sup{(μ(E))p/qcap(E,hpl): E⊂ℝn, cap(E,hpl)>0}<∞.

Here hpl(ℝn) is the completion of C0∞(ℝn) with respect to ∥f∥hpl(ℝn):=∥(−Δ)1/2f∥Lp(ℝn) , and cap(E,hpl) is the capacity of E associated with hpl(ℝn) . Such embeddings like (1.3) are referred to as trace inequalities (cf. [3]). Meanwhile, (1.4) is called the isocapacitary inequality, see Maz’ya [16]. Since the pioneer work of Maz’ya in [14], other equivalent conditions of trace inequalities were established in [1, 3, 6, 15, 17, 20]. Especially, when 1 < p < q < ∞, E in (1.3) can be replaced by balls, see [3, Theorem 7.2.2]. When 0 < q < p and 1 < p < n/l, (1.3) holds if and only if the Wolff potential Wαμ(⋅)∈Lq(p−1)/(p−q)(μ), , see Cascante-Ortega-Verbitsky [6]. There exist other conditions involving no capacity, which are equivalent to (1.3), see, e.g., [17, 20]. These equivalent conditions were widely applied to harmonic analysis, the operator theory, function spaces, linear and nonlinear partial differential equations.

When extending a function φ from ℝⁿ to ℝ+n+1 through PDEs, Xiao in [21] studied the Carleson measure characterization of embeddings of Sobolev spaces:

(1.5) ∥u(x,t2)∥Lq(ℝ+n+1,μ)≲∥∇φ∥Lp(ℝn),

where u(x, t) is the solution to the classical heat equation with the initial data φ(x). In [22], Xiao established the Carleson characterization of embeddings similar to (1.5) for the fractional Besov space Λ˙α1,1(ℝn). . On the other hand, Xiao [22] also proved that Λ˙α1,1(ℝn) embeds the Choquet space L1(H˜n−α). . In [23], Xiao proved that some isocapacitary inequalities are equivalent to strong versions of classical Sobolev inequalities. In [24], Xiao established the equivalence of the fractional version of (1.3) to several geometric inequalities in terms of fractional capacities. Motivated by Xiao [21, 22], for the solution of the spatial fractional heat equations, Zhai in [25] provided the Carleson characterization of embeddings similar to (1.5) for fractional Sobolev spaces by fractional capacities. The Carleson characterization of embeddings similar to (1.5) for Lebesgue spaces was established by Chang-Xiao in [8] and Shi-Xiao in [18] using L^p−capacities. For the fractional Poisson extension, Li et al. [13] established the Carleson characterization for the embeddings of fractional Sobolev spaces and Lebesgue spaces via fractional capacities and L^p–capacities.

Motivated by the above mentioned works, in this paper, we provide the Carleson characterization of the Lq(ℝ+n+1,μ) -extension of Sobolev spaces and Lebesgue spaces via the solution to (1.1). For equation (1.1), the solution can be written as

u(x,t)=Rα,β(φ)(x,t):=∫ℝnGt(x−y)φ(y)dy,

where G_t(·) is the space-time fractional heat kernel. Recently, there has been an increasing interest in the fractional calculus. This is because time fractional operators are proving to be very useful for modeling purposes. For example, while the classical heat equation ∂_tu(x, t) = ∆u(x, t) is used for modeling heat diffusion in the homogeneous media, the fractional heat equations

(1.6) ∂tβu(x,t)=Δu(x,t)

are used to describe heat propagation in the inhomogeneous media. It is known that as opposed to the classical heat equation, the equation (1.6) is known to exhibit the sub diffusive behaviour and is related with anomalous diffusion or diffusion in the non-homogeneous media, with random fractal structures.

Definition 1.1

Let γ ∈ (0, n) and p ∈ [1, n/γ]. The homogeneous Sobolev space Ẇ^γ,^p(ℝⁿ) is the completion of C0∞(ℝn) with respect to the norm

∥φ∥W˙γ,p(ℝn):={∥(−Δ)γ/2φ∥Lp(ℝn),p∈(1,n/γ);(∫ℝn∥Δhkφ∥Lp(ℝn)p|h|n+pγdh)1/p,p=1orp=n/γ, γ∈(0,n),

where k = 1 + [γ], γ = [γ] + {γ} with [γ] ∈ 𝕑₊, {γ} ∈ (0, 1) and

Δhkφ(x)={Δh1Δhk−1φ(x),k>1;φ(x+h)−φ(x),k=1.

Denote by T(O) the tent based on an open subset O of ℝⁿ, i.e.,

T(O)={(x,r)∈ℝ+n+1: B(x,r)⊆O}

with B(x, r) being the open ball centered at x ∈ ℝⁿ with radius r > 0. The fractional Sobolev capacities are defined as follows.

Definition 1.2

Let γ ∈ (0, n) and p ∈ [1, n/γ].

The fractional capacity of an arbitrary set S ⊂ ℝⁿ, denoted by Capℝn(γ,p)(S) , is defined as
Capℝn(γ,p)(S):=inf{∥φ∥W˙γ,p(ℝn)p: φ∈C0∞(ℝn)& φ≥1S},

where 1_S denotes the characteristic function of S.
For t ∈ (0, ∞), the (p, γ)−fractional capacity minimizing function associated with both Ẇ^γ,p(ℝⁿ) and a nonnegative measure µ on ℝ+n+1 , denoted by cpγ(μ,t) , is defined as
cpγ(μ,t):=inf{Capℝn(γ,p)(O): boundedopenO⊆ℝn, μ(T(O))>t}.

In Lemma 2.5, we provide four standard estimates involving the capacity Capℝn(γ,p)(⋅) , measures and non-tangential maximal functions. Based on Lemma 2.2, we obtain an elementary Riesz integral upper estimate of the space-time fractional heat kernel G_t(·), see Lemma 2.6. In Section 3, given α > n, β ∈ (0, 1) and a nonnegative Radon measure µ on ℝ+n+1 , we establish several characterizations of the extension:

(1.7) ∥Rα,βφ(x,tα/β)∥Lq(ℝ+n+1,μ)≲∥φ∥W˙γ,p(ℝn)

for 0 < γ < n, 1 ≤ p < n/γ and 1 < q < ∞. Our results exhibit that (1.7) holds if and only if

(1.8) {supt>0tp/qcpγ(μ;t)<∞,p≤q;∫0∞(tp/qcpγ(μ;t))q/(p−q)dtt<∞,p>q,

see Theorems 3.1, 3.2, 3.5 and 3.7, respectively. Specially, for γ = 1, our results obtained in Section 3 characterize the Lq(ℝ+n+1,μ) -extension via the p-variational capacity and generalize [21, Theorem 1.1 & 1.2], see Theorems 3.4 & 3.6.

However, it can be seen from Lemma 2.6 that the above characterizations (1.8) are invalid for the case Ẇ⁰,^p(ℝⁿ), i.e., Lebesgue spaces L^p(ℝⁿ). To investigate the extension:

Rα,β:Lp(ℝn)↦Lq(ℝ+n+1,μ)

with µ being a nonnegative Radon measure on ℝ+n+1 . We introduce the L^p–capacity related to R_α,β which is defined as follows.

Definition 1.3

Let 1 ≤ p < ∞. For any subset K⊆ℝ+n+1 , one defines

Cp(α,β)(E):=inf{∥f∥Lp(ℝn)p: f≥0 & Rα,βf≥1E}.

In Section 4, we study some fundamental properties of the Lp(ℝ+n+1) -capacity Cp(α,β)(⋅) , and further, estimate the capacities of fractional parabolic balls Br0(α,β)(x0,t0) , see Proposition 4.3. The strong and weak type inequalities corresponding to Cp(α,β)(⋅) are also derived in Lemma 4.8. Finally, in Section 5, for λ > 0, define

κ(μ;λ):=inf{Cp(α,β)(K): compactK⊂ℝ+n+1, μ(K)≥λ}.

We prove that the extension

Rα,β:Lp(ℝn)→Lq(ℝ+n+1,μ)

is bounded if and only if

(1.9) { supλ∈(0,∞)λp/qκ(μ;λ)<∞, 1<p≤q<∞;∫0∞(λp/qκ(μ;λ))q/(p−q)dλλ<∞, 1<q<p<∞,

see Theorems 5.1 and 5.2, respectively.

We point out that the characterizations obtained in Sections 3 & 5 exhibit the relation between the differential property of functions and the order of capacities. Let 1 < p < q < ∞. For the case Ẇ^γ,^p(ℝⁿ), we prove in Theorem 3.1 that one of equivalent conditions of (1.7) is

supopenO⊂ℝn(μ(T(O)))p/qCapℝnγ(O)<∞.

Specially, letting O=B(x0,r0β) the ball centered at x₀ with radius r0β , then

Capℝnγ(B(x0,r0β))≃r0β(n−γ).

On the other hand, for the case L^p(ℝⁿ), Theorem 5.1 shows that, under the assumption that t₀ ≲ r^α, the extension

∥Rα,β(φ)∥Lq(ℝ+n+1,μ)≲∥φ∥Lp(ℝn)

holds if and only if

(μ(Br(α,β)(x0,t0)))p/q≲rβn, ∀ (r,t0,x0)∈ℝ+×ℝ+×ℝn.

The above analysis indicates that, in the Lq(ℝ+n+1,μ) -extensions of Sobolev / Lebesgue spaces via (1.1), the loss of derivatives of functions need to be compensated via enhancing the order of capacities.

Some notations:

Let Ω ⊆ ℝⁿ. Throughout this article, we use C(Ω) to denote the spaces of all continuous functions on Ω. Let k ∈ ℕ₊ ∪ {∞}. The symbol C^k(Ω) denotes the class of all functions f : Ω → ℝ with k continuous partial derivatives. Let C0∞(Ω) stand for all infinitely smooth functions with compact support in Ω.
For 1 ≤ p ≤ ∞, denote by p′ the conjugate number of p, i.e., 1/p + 1/p′ = 1. U ≃ V represents that there is a constant c > 0 such that c⁻¹V ≤ U ≤ cV whose right inequality is also written as U ≲ V. Similarly, one writes V ≳ U for V ≥ cU.
For convenience, the positive constant C may change from one line to another and usually depends on the dimension n, α, β and other fixed parameters. For f ∈ 𝒮 (ℝⁿ), f^ means the Fourier transform of f.

2 Preliminaries on fractional heat kernels and Sobolev capacities

We first state some preliminaries on the space-time fractional heat kernel which will be used in the sequel. Let X_t denote a symmetric α stable process with density function denote by K_α_,_t(·). This is characterized through the Fourier transform which is given by

Kα,t^(ξ)=e−νt|ξ|α.

Let D = {D_r, r ≥ 0} denote a β-stable subordinator and E_t be its first passage time. It is known that the density of the time changed X_{E_t} is given by the G_t(x). By conditioning, we have

(2.1) Gt(x)=∫0∞Kα,t(s,x)fEt(s)ds,

where

fEt(x)=tβ−1x−1−1/βgβ(tx−1/β),

where g_β(·) is the density function of D₁, and is infinitely differentiable on the entire real line, with g_β(u) = 0 for u ≤ 0. Moreover,

gβ(u)∼K(β/u)(1−β/2)/(1−β)exp{−|1−β|(u/β)β/(β−1)}as u→0+,

and

gβ(u)∼βΓ(1−β)u−β−1as u→∞.

While the above expressions will be very important, we will also need the Fourier transform of G_t(x):

Gt^(ξ)=Eβ(−νtβ|ξ|α),

where

Eβ(x)=∑k=0∞xkΓ(1+βk)

and

11+Γ(1−β)≤Eβ(−x)≤11+Γ(1+β)−1xfor x>0.

Even though, we will be mainly using the representation given by (2.1), we also have another explicit description of the heat kernel. Using the convention ~ to denote the Laplace transform, we get

Gt^˜(x)=λβ−1λβ+ν|ξ|α.

Inverting the Laplace transform yields

Gt^(ξ)=Eβ(−ν|ξ|βtβ),

where

Eβ(x)=∑k=0∞xkΓ(1+βk),

is the Mittag-Leffler function. In order to invert the Fourier transform, we will make use of the integral

∫0∞cos(ks)Eβ,α(−asμ)ds=πkH3,32,1[kμa|(1,μ),(1,1),(1,μ/2)(1,1),(α,β),(1,μ/2)],

where ℜ(α) > 0, β > 0, k > 0, a > 0, Hp,qm,n is the H-function given in [10, Definition 1.9.1, page 55] and the formula

12π∫−∞∞e−iξxf(ξ)dξ=1π∫0∞f(ξ)cos(ξx)dξ.

This gives the function as

Gt(x)=1|x|H3,32,1[|x|ανtβ|(1,α),(1,1),(1,α/2)(1,1),(1,β),(1,α/2)].

Remark 2.1

Specially, taking α = 2, we can get, by the reduction formula for the H-function,

Gt(x)={1|x|H1,11,0[|x|2νtβ|(1,2)(1,β)],β∈(0,1);1(4νπt)1/2exp(−|x|24νt),β=1.

In [11], M. Foondun and E. Nane obtained the following estimate for G_t(·).

Proposition 2.2

([11, Lemma 2.1])

There exists a positive constant C₁ such that for all x ∈ ℝⁿ
Gt(x)≥C1min{t−βn/α, tβ|x|n+α}.
If we further suppose that α > n, then there exists a positive constant C₂ such that for all x ∈ ℝⁿ,
Gt(x)≤C2min{t−βn/α, tβ|x|n+α}.

Below we always assume that α > n. It can be deduced from (i) of Lemma 2.2 that

(2.2) Gt(x)∼tβ(|x|+tβ/α)n+α.

Also, a direct computation, together with change of variables, gives

(2.3) ∫ℝnGt(x)dx∼∫ℝntβ(|x|+tβ/α)n+αdx≲1.

The following estimate is an immediate corollary of Lemma 2.2.

Lemma 2.3

Let 1 ≤ r ≤ p ≤ ∞ and φ ∈ L^r(ℝⁿ). For α > n & t > 0, we have

∥Rα,β(φ)∥Lp(ℝn)≲t−nβ(1/r−1/p)/α∥φ∥Lr(ℝn).

Proof

Let q obey 1/r + 1/q = 1/p + 1. By Young’s inequality,

∥Rα,β(φ)∥Lp(ℝn)=∥Gt*φ∥Lp(ℝn)≲∥φ∥Lr(ℝn)∥Gt(⋅)∥Lq(ℝn).

Applying Lemma 2.2 and a direct computation, we get

∥Gt(⋅)∥Lq(ℝn)∼(∫ℝntβq(|x|+tβ/α)q(n+α)dx)1/q≲tβn(1/q−1)/α,

which implies that

∥Rα,β(φ)∥Lp(ℝn)≲tβn(1/q−1)/α∥φ∥Lr(ℝn)=t−βn(1/r−1/p)/α∥φ∥Lr(ℝn).

Below we state some basic properties of the capacities Capℝn(γ,p)(⋅) and refer the reader to [3, Secton 2] for the details.

Proposition 2.4

The following properties are valid.

Capℝn(γ,p)(∅)=0 .
If K₁ ⊆ K₂ ⊂ ℝⁿ, then Cap ℝn(γ,p)(K1)≤Cap ℝn(γ,p)(K2) .
For any sequence {Kj}j=1∞ of subsets of ℝ+n+1 ,
Capℝn(γ,p)(∪j=1∞Kj)≤∑j=1∞Capℝn(γ,p)(Kj).
For any decreasing sequence {Kj}j=1∞ in ℝⁿ with K = ∩_jK_j and any increasing sequence {Ej}j=1∞ in ℝⁿ with E = ∪_jE_j, one has
{ limj→∞Cap ℝn(γ,p)(Kj)=Cap ℝn(γ,p)(K),limj→∞Cap ℝn(γ,p)(Ej)=Cap ℝn(γ,p)(E).

Let 𝔐+(ℝ+n+1) represent the class of all nonnegative Radon measures on ℝ+n+1 .

Lemma 2.5

Let α > n, β ∈ (0, 1) and γ ∈ (0, n). Given φ ∈ Ẇ^γ,p(ℝⁿ), p ≥ 1_, s > 0, and μ∈𝔐+(ℝ+n+1) , define

Lsα,β(φ):={(x,t)∈ℝ+n+1: |Rα,βφ(x,tα/β)|>s}

and

Msα,β(φ):={y∈ℝn: sup|y−x|<t|Rα,βφ(x,tα/β)|>s}.

Then the following four statements are true.

For any natural number k
μ(Lsα,β(φ)∩T(B(0,k)))≤μ(T(Msα,β(φ)∩B(0,k))).
For any natural number k,
Capℝnγ,p(Msα,β(φ)∩B(0,k))≥cpγ(μ,μ(T(Msα,β(φ)∩B(0,k)))).
There exists a constant θ_n_,_α > 0 such that
sup|y−x|<t|Rα,βφ(y,tα/β)|≤θn,α𝔐φ(x), x∈ℝn,
where 𝔐 denotes the Hardy-Littlewood maximal operator:
𝔐φ(x)=supr>0r−n∫B(x,r)|φ(y)|dy, x∈ℝn.
Let O be a bounded open set contained in Int ({x ∈ ℝⁿ : φ(x) ≥ 1}). There exists a constant η_n,α > 0 such that if (x, t) ∈ T(O), then R_α,β(|φ|)(x, t^α/β) ≥ η_n_,_α.

Proof

Since sup_|_y₋_x_|<_t |R_α_,_βφ(x, t^α/β)| is lower semicontinuous on ℝⁿ, we can see that Msα,β(φ) is an open subset of ℝⁿ and
{Lsα,β(φ)⊆T(Msα,β(φ));μ(Lsα,β(φ))≤μ(T(Msα,β(φ))).

Then
μ(Lsα,β(φ)∩T(B(0,k)))≤μ(T(Msα,β(φ)∩T(B(0,k))))=μ(T(Msα,β(φ)∩B(0,k))).
It follows from the definition of cpγ(μ;t) .
Let Φ_t(x) := t ⁻ⁿΦ(x/t) with Φ(x) = (|x| + 1)⁻ⁿ⁻^α. By (2.2), it holds
sup|y−x|<t|Rα,βφ(y,tα/β)|≤C1sup|y−x|<t|Φt*φ(y)|.

Since Φ is radial, bounded, decreasing and integrable on ℝⁿ, it follows from [19, page 57, Proposition] that
sup|y−x|<t| Φt*φ(y) |≤C2𝔐φ(x)
and thus for a constant θ_n_,_α,
sup|y−x|<t|Rα,βφ(y,tα/β)|≤θn,α𝔐φ(x).
For any (x, t) ∈ T(O), we have
B(x,t)⊆O⊆ Int ({x:φ(x)>1}).

It can be deduced from (i) of Proposition 2.2 that there exist σ and C such that
inf{Gtα/β(x):|x|<σt}≥Ct−n.

Then
Gtα/β*|φ|(x,t)≥Ct−n∫B(x,σt)∩Int({x: φ(x)≥1})|φ|(y)dy.

If σ > 1, then
B(x,σt)∩Int({x:φ(x)≥1})⊇B(x,t)∩Int({x:φ(x)≥1})=B(x,t).

If σ ≤ 1, then
B(x,σt)∩Int({x:φ(x)≥1})=B(x,σt).

Thus G_tα/β ★ |φ|(x, t) ≥ η_n,α for some constant η_n,α > 0.

Lemma 2.6

If α > n, β ∈ (0, 1), γ ∈ (0, n) and (x,t)∈ℝ+n+1, , then

∫ℝnGtα/β(y)|y−x|n−γdy≲(t+|x|)γ−n.

Proof

Note that Gt(x)∼tβ(|x|+tβ/α)n+α. . Define

J(x,t)=∫ℝntα|y−x|γ−n(|x|+t)n+αdy.

Via the change of variables: x −→ tx & y −→ ty, it is sufficient to show that

J(x,1)≲(1+|x|)γ−n.

Since J(0, 1) ≲ 1, we may assume that |x| > 0. Write J(x, 1) ≲ I₁(x) + I₂(x), where

{I1(x):=∫B(x,|x|/2)|y−x|γ−n(|x|+1)n+αdy;I2(x):=∫ℝn\B(x,|x|/2)|y−x|γ−n(|x|+1)n+αdy.

Since |x − y| ≤ |x|/2 implies that |y| ≃ |x|, we have

I1(x)≲(1+|x|)−(n+α)∫B(x,|x|/2)|y−x|γ−ndy≲(1+|x|2)−(n+α)/2∫0|x|/2sγ−1ds≲(1+|x|2)−(n+α)/2|x|γ≲(1+|x|)γ−n.

If |x − y| > |x|/2, then

I2(x)≲|x|γ−n∫ℝn\B(x,|x|/2)1(|y|+1)(n+α)dy.

For |x − y| > |x|/2, it holds |y| < 3|x − y| and

I2(x)≲∫ℝn\B(x,|x|/2)1(|y|+1)n+α|y|n−γdy≲1.

So, I₂(x) ≲ (1 + |x|)^γ⁻ⁿ and J(x, 1) ≲ (1 + |x|)^γ−n.

The following result provides the capacitary strong estimates for Capℝn(γ,p)(⋅) . For the proofs, we refer the readers to [4, 22] and the references therein.

Lemma 2.7

Let γ ∈ (0, n) and p ∈ [1, n/γ].

For φ∈C0∞(ℝn) ,
∫0∞Capℝn(γ,p)({x∈ℝn:|f(x)|≥s})dsp≲∥φ∥W˙γ,p(ℝn)p.
For φ∈C0∞(ℝn) ,
∫0∞Capℝn(γ,p)({x∈ℝn:|𝔐φ(x)|≥s})dsp≲∥φ∥W˙γ,p(ℝn)p.

For handling the endpoint case p = nγ, we need the following Riesz potentials on ℝ²ⁿ, see Adam-Xiao [4] and Adam [2]. For γ ∈ (0, 2n),

Iγ(2n)*f(z):=∫ℝ2n|x−y|γ−2nf(y)dy, z∈ℝ2n.

For γ ∈ (0, 2n), the space ℒ˙γp(ℝ2n) is defined as the set

{f: f=Iγ(2n)*φ, φ∈Lp(ℝ2n)}

with the norm ∥f∥ℒ˙γp(ℝ2n):=∥Iγ(2n)*φ∥ℒ˙γp(ℝ2n)=∥φ∥Lp(ℝ2n) . Formally, we write ℒ˙γp(ℝ2n)=Iγ(2n)*Lp(ℝ2n) .

The following result is a particular case of [2, Theorem 5.2. or [4, Theorem A].

Lemma 2.8

Let γ ∈ (0, n). There are a linear extension operator

ℰ: W˙γ,n/γ(ℝn)→ℒ˙2γn/γ(ℝ2n),

and a linear restriction operator

ℛ: ℒ˙2γn/γ(ℝ2n)→W˙γ,n/γ(ℝn)

such that ℜ𝔈 is the identity. Moreover,

For φ ∈ Ẇ^γ,n/γ(ℝⁿ), ∥ℰf∥ℒ˙2γnγ(ℝ2n)≲∥φ∥W˙γ,n/γ(ℝn) .
For g∈ℒ˙2γn/γ(ℝ2n) , ∥ℛg∥W˙γ,n/γ(ℝn)≲∥g∥ℒ˙2γn/γ(ℝ2n) .

3 Lq(ℝ+n+1,μ) −extension of Ẇ^γ,p(ℝⁿ)

In this section, we will show that the embedding (1.7) can be characterized by conditions in terms of fractional capacities. For 0 < p, q < ∞ and a nonnegative Radon measure µ on ℝ+n+1 , Lq,p(ℝ+n+1,μ) and Lq(ℝ +n+1,μ) denote the Lorentz space and the Lebesgue space of all functions on ℝ +n+1 , respectively, for which

∥g∥Lq,p(ℝ +n+1,μ):={ ∫0∞(μ({ (x,t)∈ℝ +n+1: |g(x,t)|>s }))p/qdsp }1/p<∞

and

∥g∥Lq(ℝ +n+1,μ):=(∫ℝ +n+1|g(x,t)|qdμ)1/q<∞,

respectively. Moreover, we denote by Lq,∞(ℝ +n+1,μ) the set of all µ–measurable functions g on ℝ +n+1 with

∥g∥Lq,∞(ℝ +n+1,μ):=sups>0s(μ({ (x,t)∈ℝ +n+1:|g(x,t)|>s }))1/q<∞.

3.1 Lq(ℝ +n+1,μ) −extension of Ẇ^γ,p(ℝⁿ) when p ≤ q

Theorem 3.1

Let α > n, β ∈ (0, 1). Let γ ∈ (0, n) when 1 ≤ p ≤ n/γ, p ≤ q < ∞ and μ∈𝔐+(ℝ +n+1) . Then the following statements are equivalent:

∥Rα,βφ(x,tα/β)∥Lq,p(ℝ +n+1,μ)≲∥φ∥W˙γ,p(ℝn) ∀φ∈W˙γ,p(ℝn) ;
∥Rα,βφ(x,tα/β)∥Lq(ℝ +n+1,μ)≲∥φ∥W˙γ,p(ℝn) ∀φ∈W˙γ,p(ℝn) ;
∥Rα,βφ(x,tα/β)∥Lq,∞(ℝ +n+1,μ)≲∥φ∥W˙γ,p(ℝn) ∀φ∈W˙γ,p(ℝn) ;
supt>0tp/qc pγ(μ;t)<∞ ;
(μ(T(O)))p/q≲Cap ℝnγ,p(O) holds for any bounded open set O ⊆ ℝⁿ.

Proof

The implications (i) ⇒ (ii) ⇒ (iii) can be deduced from

(sqμ(L sα,β(φ)))p/q≤(q∫0∞μ(L sα,β(φ))sq−1ds)p/q≤∫0∞(μ(L sα,β(φ)))p/qdsp

since

qμ(L rα,β(φ))rq−1≤ddr(∫0r(μ(L sα,β(φ)))p/qdsp)q/p.

Now, we prove (iii) ⇒ (v) ⇒ (i). If (iii) is true, i.e.,

Kp,q(μ)=supφ∈C 0∞(ℝn)&∥φ∥W˙γ,p(ℝn)>0sups>0s(μ({ (x,t)∈ℝ +n+1: |Rα,βφ(x,tα/β)|>s }))1/q∥φ∥W˙γ,p(ℝn)<∞,

then Lemma 2.5 implies

(3.1) (μ(T(O)))1/q≲Kp,q(μ)∥φ∥W˙γ,p(ℝn)

for any f∈C 0∞(ℝn) and any open set O ⊆ Int ({x ∈ ℝⁿ : f(x) ≥ 1}). Thus (v) holds. For (v) ⇒ (i), denote

Qp,q(μ):=sup{ (μ(T(O)))p/qCap ℝn(γ,p)(O): boundedopenset O⊆ℝn }<∞.

Lemmas 2.5 & 2.7 imply that

∫0∞(μ(L sα,β(φ)∩T(B(0,k))))p/qdsp ≤∫0∞(μ(T(M sα,β(φ)∩B(0,k))))p/qdsp ≤∫0∞(μ(T({x∈ℝn:θn,α𝔐(φ)(x)>s}∩B(0,k))))p/qdsp ≤Qp,q(μ)∫0∞Cap ℝn(γ,p)({x∈ℝn:θn,α𝔐(φ)(x)>s}∩B(0,k))dsp ≤Qp,q(μ)∫0∞Cap ℝn(γ,p)({x∈ℝn:θn,α𝔐(φ)(x)>s})dsp ≤Qp,q(μ)∥φ∥ W˙γ,p(ℝn)p

for any φ∈C 0∞(ℝn) . Letting k → ∞ reaches (i).

Now, we will prove (iii) ⇒ (iv) ⇒ (i). If (iii) is true, let O be a bounded open subset of Int({x ∈ ℝⁿ, φ(x) ≥ 1}). Then

tp/q≲(Kp,q(μ))pCap ℝn(γ,p)(O)

whenever t ∈ (0, µ(T(O))). So, tp/q≲(Kp,q(μ))pc pγ(μ;t) , which means (iv) is true.

Assume that (iv) is true. By Lemmas 2.5 & 2.7, for any φ∈C 0∞(ℝn) ,

∫0∞(μ(L sα,β(φ)∩T(B(0,k))))p/qdsp ≤∫0∞(μ(L sα,β(φ)∩T(B(0,k))))p/qc pγ(μ;μ(L sα,β(φ)∩T(B(0,k))))Cap ℝn(γ,p)(M sα,β(φ)∩B(0,k))dsp ≲supt>0tp/qc pγ(μ;t)∫0∞Cap ℝn(γ,p)({x∈ℝn:θn,α𝔐(φ)(x)>s}∩B(0,k))dsp ≲supt>0tp/qc pγ(μ;t)∥φ∥ W˙γ,p(ℝn)p,

which gives (i) via letting k → ∞.

Theorem 3.2

Let μ∈𝔐+(ℝ +n+1) , α > n, β ∈ (0, 1), γ ∈ (0, n), 1 < p < min{n/γ, q}, or 1 = p ≤ q < ∞. Then the following two statements are equivalent.

For any φ ∈ Ẇ^γ,p(ℝⁿ),
(∫ℝ +n+1| Rα,βφ(x,tα/β) |qdμ(x,t))1/q≲∥φ∥W˙γ,p(ℝn).
supx∈ℝn,r>0(μ(T(B(x,r))))p/qCap ℝn(γ,p)(B(x,r))<∞.

Proof

It is enough to prove that (ii) implies (iii) or (v) of Theorem 3.1. When 1 = p ≤ q < ∞, if (ii) holds, then

∥μ∥1,q=supx∈ℝn,r>0(μ(T(B(x,r))))1/qCap ℝn(γ,1)(B(x,r))<∞.

Suppose that a bounded open set O ⊆ ℝⁿ is covered by a sequence of dyadic cubes {I_j} in ℝⁿ with ∑j|Ij|(n−γ)/n<∞ . It follows from [9, Lemma 4.1] that there exists another sequence of dyadic cubes {J_j} in ℝⁿ such that

{ Int(Jj)∩Int(Ik)=∅, j≠k;∪kIk=∪jJj;∑j|Jj|1−γ/n≤∑k|Ik|1−γ/n;T(O)≲∪jT(Int(5nJj)).

Thus,

μ(T(O))≲∥μ∥1,q∑j|5nJj|q(n−γ)/n≲∥μ∥1,q(∑k|Ik|(n−γ)/n)q,

which implies

μ(T(O))≲∥μ∥1,qH ∞n−γ(O).

Here H ∞d(⋅) is the d–dimensional Hausdorff capacity. Then, the classical result Cap ℝn(γ,1)(⋅)∼H ∞d(⋅) implies

μ(T(O))≲∥μ∥1,q(Cap ℝn(γ,1)(O))q.

Thus, (v) of Proposition 3.1 holds.

When 1 < p < min{n/γ, q},

∥μ∥p,q=supx∈ℝn,r>0(μ(T(B(x,r))))p/qCap ℝn(γ,p)(B(x,r))<∞.

Let φ₀ ∈ Ẇ^γ,p(ℝⁿ and µ_s be the restriction of µ to L sα,β(W˙γ,p(ℝn)) . For any φ ∈ Ẇ^γ,p(ℝⁿ), we have

|φ(x)|≲∫ℝn|(−Δ)γ/2φ(y)||x−y|n−γdy, x∈ℝn.

Thus, Lemma 2.6 implies

sμ(L sα,β(φ0))≲∫L sα,β(φ0)| Rα,βφ0(x,tα/β) |dμ(x,t)≲∫L sα,β(φ0)| ∫ℝnGtα/β(y)φ0(x−y)dy |dμ(x,t)≲∫L sα,β(φ0){ ∫ℝnGtα/β(y)(∫ℝn| (−Δ)γ/2φ0(z)) ||x−y−z|n−γdz)dy }dμ(x,t)≲∫L sα,β(φ0){ ∫ℝn(∫ℝnGtα/β(y)dy|y−(x−z)|n−γ)| (−Δ)γ/2φ0(z) |dz }dμs(x,t)≲∫ℝ +n+1{ ∫ℝn| (−Δ)γ/2φ0(z) |(t+|x−z|)n−γdz }dμs(x,t)≲∫ℝn| (−Δ)γ/2φ0(z) |{ ∫ℝ +n+1dμs(x,t)(t+|x−z|)n−γ }dz≲∫ℝn| (−Δ)γ/2φ0(z) |{ ∫0∞μs(T(B(z,r))rγ−n−1dr) }dz≲J(λ)+K(λ),

where

J(λ):=∫0λ{ ∫ℝn| (−Δ)γ/2φ0(z) |μs(T(B(z,r))dz }rγ−n−1dr

and

K(λ):=∫λ∞(∫ℝn| (−Δ)γ/2φ0(z) |μs(T(B(z,r))dz)rγ−n−1dr.

Since Cap ℝn(γ,p)(B(x,r))∼rn−pγ , it follows from the definition of ‖µ‖_p,q and Hölder’s inequality that for 1/p + 1/p′ = 1,

μs(T(B(z,r)))≲(μs(T(B(z,r))))1/p′∥μ∥ p,qq/p2rq(n−pγ)/p2.

Thus we have

∫ℝnμs(TB(x,r))dx≲rnμs(L sα,β(φ0))

and

J(λ)≲∫0λ{ ∫ℝn| (−Δ)γ/2φ0(z) |(μs(T(B(z,r))))1/p′∥μ∥ p,qq/p2rq(n−pγ)/p2dz }rγ−n−1dr≲∥φ0∥W˙γ,p(ℝn)∥μ∥ p,qq/p2∫0λ{ ∫ℝnμs(T(B(z,r)))dz }1/p′rq(n−pγ)/p2+γ−n−1dr≲∥φ0∥W˙γ,p(ℝn)∥μ∥ p,qq/p2∫0λ(rnμs(T(B(z,r)))1/p′rq(n−pγ)/p2+γ−n−1dr≲∥φ0∥W˙γ,p(ℝn)∥μ∥ p,qq/p2(μ(L sα,β(φ0))1/p′λ(q−p)(n−pγ)/p2.

Similarly, we can obtain

K(λ)≲∥φ0∥W˙γ,pμ(L sα,β(φ0)λγ−n/p.

Thus,

(3.2) sμ(Lsα,β(φ0)≲∥φ0(⋅,tα/β)∥W˙γ,pμs(Lsα,β(φ0))×{ λγ−n/p+∥μ∥p,qq/p2(μs(Lsα,β(φ0)))−1/pλ(q−p)(n−pγ)/p2 }.

In (3.2), take

λ=∥μ∥ p,q1/(pγ−n)(μs(L sα,β(φ0)))pq(n−pγ)

such that

λγ−n/p=∥μ∥ p,qq/p2(μs(L sα,β(φ0)))−1/pλ(q−p)(n−pγ)/p2.

It holds

s(μ(L sα,β(φ0))1/q≲∥μ∥ p,q1/p∥φ0∥W˙γ,p(ℝn)

which implies (iii) of Proposition 3.1.

Let q = np/(n − pγ) and µ(x, t) = δ(x₀,t₀) be the Dirac measure at (x₀, t₀). If (x₀, t₀) ∈ T(B(x, r)), then δ(x₀,t₀)(T(B(x, r)) = 0. If (x₀, t₀) ∈ T(B(x, r)), then B(x₀, t₀) ⊆ B(x, r) and t 0n≤rn and so

δ(x0,t0)(T(B(x,r)))≤rnt 0n=r(n−pγ)q/pt 0n.

Then the equivalence Cap ℝn(γ,p)(B(x,r))∼rn−pγ implies

δ(x0,t0)T(B(x,r))p/qCap ℝn(γ,p)(B(x,r))≤t 0−np/q.

So,

∥δ(x0,t0)(T(B(x,r)))∥p,q=t 0pγ−n.

It follows from the proof of Proposition 3.2 that

|Rα,βφ(x0,t 0α/β)|≲∥δ(x0,t0)∥p,q∥φ∥W˙γ,p(ℝn).

Thus, we have the following decay estimate of the solution.

Corollary 3.3

Let α > n and β ∈ (0, 1). If φ₀ ∈ Ẇ^γ,p(ℝⁿ, then for 1 ≤ p < n/γ and γ ∈ (0, n),

|Rα,βφ(x0,t0)|≲t 0β(pγ−n)/α∥φ∥W˙γ,p(ℝn) ∀ (x0,t0)∈ℝ +n+1.

Let cap_p(S) denote the p-variational capacity of an arbitrary set S ⊆ ℝⁿ defined by

capp(S):=inf{ ∫ℝn|∇f(x)|pdx: f∈W˙1,p(ℝn), S⊆Int({x∈ℝn: f(x)≥1}) },

where Int (E) stands for the interior of a set E ⊆ ℝⁿ; c_p(µ, t) denotes the corresponding p-variational capacity minimizing function of t ∈ (0, ∞) associated with a nonnegative measure µ on ℝ +n+1 :

cp(μ,t):=inf{capp(O): boundedopenO⊆ℝn, μ(T(O))>t}.

As an immediate corollary of Theorem 3.1. it holds

Theorem 3.4

Let α > n, β ∈ (0, 1). Assume that p ≤ q < ∞, 1 ≤ p < n and μ∈𝔐+(ℝ +n+1). . The following statements are equivalent:

(∫ℝ +n+1| Rα,βφ(x,tα/β) |qdμ(x,t))1/q≲∥∇φ∥Lp(ℝn) ∀ φ∈W˙1,p(ℝn);
supλ>0λ(μ({ (x,t)∈ℝ +n+1: | Rα,βφ(x,tα/β) |>λ }))1/q≲∥∇φ∥Lp(ℝn) ∀ φ∈W˙γ,p(ℝn);
supt>0tp/qcp(μ;t)<∞;
sup{ (μ(T(O)))p/qcapp(O): boundedopenO⊆ℝn }.

3.2 Lq(ℝ +n+1,μ) −extension of Ẇ^γ,p(ℝⁿ) when p > q

Theorem 3.5

Let α > n, β ∈ (0, 1), γ ∈ (0, n), 0 < q < p, 1 < p ≤ n/γ and μ∈𝔐+(ℝ +n+1) . Then the following statements are equivalent:

(∫ℝ +n+1| Rα,βφ(x,tα/β) |qdμ(x,t))1/q≲∥φ∥W˙γ,p(ℝn) ∀ φ∈W˙γ,p(ℝn);
(3.3) ∫0∞(tp/qc pγ(μ;t))q/(p−q)dtt<∞.

Proof

(ii) ⇒ (i). If (ii) holds, then

Ip,q(μ):=∫0∞(tp/qc pγ(μ;t))q/(p−q)dtt<∞.

For each φ∈C 0∞(ℝn) , each j = 0, ±1, ±2, . . ., and each natural number k, Lemma 2.5 (iii) implies that

Cap ℝn(γ,p)(M 2jα,β(φ)∩(B(0,k)))≤Cap ℝn(γ,p)({ x∈ℝn:θn,α𝔐φ(x)>2j }∩B(0,k)).

Define

{ μj,k(φ):=μ(T(M 2jα,β(φ)∩(B(0,k))));Sp,q,k(μ;φ):=∑j=−∞∞(μj,k(φ)−μj+1,k(φ))p/(p−q)(Cap ℝn(γ,p)(M 2jα,β(φ)∩(B(0,k))))q/(p−q).

It follows from (ii) of Lemma 2.5 that

(Sp,q,k(μ;φ))(p−q)/p≲(∑j=−∞∞μ j,kp/(p−q)(φ)−μ j+1,kp/(p−q)(φ)(c pγ(μ;μj,k(φ)))q/(p−q))(p−q)/p≲(∫0∞1(c pγ(μ;s))q/(p−q)dsp/(p−q))(p−q)/p≃(Ip,q(μ))(p−q)/p.

On the other hand, By (ii) of Lemma 2.7 and (ii)–(iii) of Lemma 2.5, we can apply Hölder’s inequality to deduce that

∫T(B(0,k))|Rα,βφ(x,tα/β)|qdμ(x,t) =∫0∞μ(L sα,β(φ)∩T(B(0,k)))dsq ≲∑j=−∞∞2jq(μj,k(φ)−μj+1,k(φ)) ≲(Sp,q,k(μ;φ))1−q/p(∑j=−∞∞2jpCap ℝn(γ,p)(M 2j(φ)α,β∩(B(0,k)))q/p ≲(Sp,q,k(μ;φ))1−q/p(∫0∞Cap ℝn(γ,p)({ x∈ℝN:θn,α𝔐φ(x)>2j }∩(B(0,k))dsp)q/p ≲(Sp,q,k(μ;φ))1−q/p∥φ∥ W˙γ,p(ℝn)q.

So, we get

(∫T(B(0,k))|Rα,βφ(x,tα/β)|qdμ(x,t))1/q≲(Ip,q(μ))(p−q)/pq∥φ∥W˙γ,p(ℝn).

Letting k → ∞ derives (1.7).

(i)⇒ (ii). If (i) holds, then

Cp,q(μ):=supφ∈C 0∞(ℝn) & ∥φ∥W˙γ,p(ℝn)>01∥φ∥W˙γ,p(ℝn)(∫ℝ +n+1|Rα,βφ(x,tα/β)|qdμ(x,t))1/q<∞.

For each φ∈C 0∞(ℝn) with ‖φ‖Ẇ_γ,p_(ℝⁿ) > 0, there holds

(∫ℝ +n+1|Rα,βφ(x,tα/β)|qdμ(x,t))1/q≤Cp,q(μ)∥φ∥W˙γ,p(ℝn),

which indicates that

sups>0s(μ(L sα,β(φ)))1/q≲Cp,q(μ)∥φ∥W˙γ,p(ℝn).

This, together with (iv) of Lemma 2.5, implies that for fixed φ∈C 0∞(ℝn) and any bounded open set O ⊆ Int({x ∈ ℝⁿ : φ(x) ≥ 1}).

μ(T(O))≲C p,qq(μ)∥φ∥ W˙γ,p(ℝn)q.

The definition of c pγ(μ;t) implies that c pγ(μ;t)>0 . For t ∈ (0, ∞) and every j, there exists a bounded open set O_j ⊆ ℝⁿ such that

{ Cap ℝn(γ,p)(Oj)≤2c pγ(μ;2j);μ(T(Oj))>2j.

Below we divide the rest of the proof into two cases.

Case 1: p ∈ (1, n/γ). Since

Capℝn(γ,p)(S)≃inf{ ∥g∥ Lp(ℝn)p:g∈C 0∞(ℝn), g≥0, S⊆ Int ({x∈ℝn:Iγ*g(x)≥1}) },

there exists gj∈C 0∞(ℝn) such that

{ Iγ*gj(x)≥1, x∈Oj;∥gj∥ Lp(ℝn)p≤2Cap ℝn(γ,p)(Oj)≤4c pγ(μ;2j).

For the integers i, k with i < k, define

gi,k=supi≤j≤k(2jc pγ(μ;2j))1/(p−q)gj.

Then g_i,k ∈ L^p(ℝⁿ) with

∥gi,k∥ Lp(ℝn)p≲∑j=ik(2jc pγ(μ;2j))p/(p−q)c pγ(μ;2j).

It is easy to see that for i ≤ j ≤ k, if x ∈ O_j, then

Iγ*gi,k(x)≥(2jc pγ(μ;2j))1/(p−q).

Set

ui,k(x,t)=Rα,β(Iγ*gi,k)(x,tα/β).

We can deduce from (iv) of Lemma 2.5 that there exists a constant η_n,α such that for (x, t) ∈ T(O_j),

|ui,k(x,t)|≥(2jc pγ(μ;2j))1/(p−q)ηn,α.

Thus, with s=(2jc pγ(μ;2j))1/(p−q)ηn,α2 ,

2j≤μ(T(Oj))≤μ(L sα,β(Iγ*gi,k(x))).

Equivalently, if s > 0 such that μ(L sα,β)(Iγ*gi,k(x))≤s , then

s>(2jc pγ(μ;2j))1/(p−q)ηn,α2.

This indicates that

(Cp,q(μ)∥gi,k∥Lp(ℝn))q≳∫ℝ +n+1|ui,k(x,tα/β)|qdμ(x,t)≃∫0∞(inf{ s:μ(L sα,β)(Iγ*gi,k(x))≤s })qds≳∑j=ik(inf{ s:μ(L sα,β)(Iγ*gi,k(x))≤2j })q2j≳∑j=ik2j(2jc pγ(μ;2j))q/(p−q)≳(∑j=ik2jp/(p−q)c pγ(μ;2j)q/(p−q))(p−q)/p∥gi,k∥ Lp(ℝn)q.

Thus,

∑j=ik2jp/(p−q)c pγ(μ;2j)q/(p−q)≲(Cp,q(μ))pq/(p−q).

Case 2: p = n/γ. The definition of Capℝn(γ,p)(⋅) implies that there exists a positive function φj∈C0∞(ℝn) such that φ_j ≥ 1 on O_j and

∥φj∥W˙γ,p(ℝn)p≤2Capℝn(γ,p)(Oj)≤4cpγ(μ;2j).

By Lemma 2.8, there exists g_i(·, ·) ∈ L^p(ℝ²ⁿ) such that

φj(x)=I2γ(2n)*gj(x,0)=ℛℰφj(x)

and

∥I2γ(2n)*gj∥ℒ˙2γp(ℝ2n)=∥ℰφj∥ℒ˙2γp(ℝ2n)≲∥φj∥W˙γ,p(ℝn).

We can define g_i,k similar to the previous case. It is easy to show that g_i,k ∈ L^p(ℝ²ⁿ) and I2γ(2n)*gi,k∈ℒ˙2γp(ℝn) . Then Lemma 2.8 implies

∥ℛI 2γ(2n)*gi,k∥ W˙γ,p(ℝn)p≲∑j=ik(2jc pγ(μ;2j))p/(p−q)∥I 2γ(2n)*gi,k∥ ℒ˙p(ℝ2n)p≲∑j=ik(2jc pγ(μ;2j))p/(p−q)∥φj∥ W˙γ,p(ℝn)p≲∑j=ik(2jc pγ(μ;2j))p/(p−q)c pγ(μ;2j).

Then consider ℛI 2γ(2n)*gi,k . Similar to the previous case, we can get

∑j=ik2jp/(p−q)c pγ(μ;2j)q/(p−q)≲(Cp,q(μ))pq/(p−q).

Letting i, k → ∞, we reach

∫0∞(tp/qc pγ(μ;t))q/(p−q)≲∑−∞∞2jp/(p−q)(c pγ(μ;2j))q/(p−q)≲(Cp,q(μ))pq/(p−q),

which implies (ii).

Similar to Theorem 3.4. letting γ = 1 in Theorem 3.5. we can obtain

Theorem 3.6

Let α > n, β ∈ (0, 1). Assume that 0 < q < p, 1 ≤ p < n and μ∈𝔐+(ℝ +n+1) . The following states are equivalent:

(∫ℝ +n+1| Rα,βφ(x,tα/β) |qdμ(x,t))1/q≲∥∇φ∥Lp(ℝn) ∀ φ∈W˙1,p(ℝn);
∫0∞(tp/qcp(μ;t))q/(p−q)dtt<∞.

When q < p = 1, we can establish the following necessary conditions for the embedding (1.7).

Proposition 3.7

Let γ ∈ (0, 1), 0 < q < p = 1 and μ∈𝔐+(ℝ +n+1) . Then (i)⇒ (ii) ⇒ (iii) ⇒ (iv):

∥Rα,βφ(x,tα/β)∥Lq(ℝ +n+1,μ)≲∥φ∥W˙γ,1(ℝn) ∀ φ∈W˙γ,1(ℝn) ;
∥Rα,βφ(x,tα/β)∥Lq,∞(ℝ +n+1,μ)≲∥φ∥W˙γ,1(ℝn) ∀φ∈W˙γ,1(ℝn) ;
sup{ (μ(T(O)))p/qCap ℝn(γ,p)(O):boundedopenO⊆ℝn }<∞ ;
∥Rα,βφ(x,tα/β)∥Lq,1(ℝ +n+1,μ)≲∥φ∥W˙γ,1(ℝn) ∀φ∈W˙γ,1(ℝn) .

Proof

The proofs of (i) ⇒ (ii) ⇒ (iii) are similar to those of (ii) ⇒ (iii) ⇒ Theorem 3.1 (v). The implication (iii) ⇒ (iv) follows from the estimate

μ(L sα,β(φ))≤μ(T(M sα,β(φ)))≤μ(T({ x∈ℝn:θn,α𝔐φ(x)>s }))≲(Cap ℝn(γ,1)({ x∈ℝn:θn,α𝔐φ(x)>s }))q,

which is a consequence of Lemma 2.5 and Theorem 3.1 (v).

4 L^p-capacity for space-time fractional heat equations

To establish the adjoint formulation of C p(α,β)(⋅) , we need to find out the adjoint operator R α,β* of the operator R_α,β. Note that

∬ℝ +n+1Rα,β(f)(x,t)g(x,t)dxdt=∬ℝ +n+1(∫ℝnGt(x−y)f(y)dy)g(x,t)dxdt=∫ℝnf(y)(∬ℝ +n+1Gt(x−y)g(x,t)dxdt)dy

holds for all f, g∈C 0∞(ℝ +n+1) . So R α,β* is defined via

R α,β*(g)(x):=∬ℝ +n+1Gt(x−y)g(x,t)dxdt, g∈C 0∞(ℝ +n+1).

For a Borel measure µ with compact support in ℝ +n+1 ,

R α,β*μ(x):=∬ℝ +n+1Gt(z−x)dμ(z,t).

Proposition 4.1

Given p ∈ (1, ∞) and a compact subset K of ℝ +n+1 , let p′ = p/(p − 1) and 𝔐₊(K) be the class of nonnegative Radon measures supported by K.

C p(α,β)(K)=sup{∥μ∥ 1p: μ∈𝔐+(K) & ∥R α,β*μ∥Lp′(ℝn)≤1} .
There exists a µ_K ∈ 𝔐₊(K) such that
μK(K)=∫ℝn(R α,β*μK(x))p′dx=∬ℝ +n+1Rα,β(R α,β*μK)p′dx=C p(α,β)(K).

Proof

(i) We set

C p(α,β)˜(K)=sup{∥μ∥ 1p: μ∈𝔐+(K) & ∥R α,β*μ∥Lp′(ℝn)≤1}.

Since ‖µ‖₁ = µ(K), for any f ≥ 0 & R_α,βf ≥ 1_K,

∥μ∥1≤∬KRα,βf(x,t)dμ(x,t)≤∬ℝ +n+1Rα,βf(x,t)dμ(x,t)≤∫ℝnf(x)R α,β*μ(x)dx≤∥f∥Lp(ℝn)∥R α,β*μ∥Lp′(ℝn).

Because ∥R α,β*μ∥Lp′(ℝn)≤1 , then ‖µ‖₁ ≤ ‖f‖_L^p(ℝⁿ) and

∥μ∥ 1p≤inf{∥f∥ Lp(ℝn)p:f≥0&Rα,βf≥1K}

which means that C p(α,β)˜(K)≤C p(α,β)(K) .

Define

{ 𝒳:={μ:μ∈𝔐+(K)&μ(K)=1};𝒴:={f:0<f∈Lp(ℝn)&∥f∥Lp(ℝn)≤1};𝒵:={f:0≤f∈Lp(ℝn)&Rα,βf≥1K};E(μ,f):=∫ℝn(R α,β*μ)(x)f(x)dx=∬ℝ +n+1Rα,βf(x,t)dμ(x,t).

By [3, Theorem 2.4.1], sup_f_∈𝒴 min_µ_∈𝒳 E(µ, f) = min_µ_∈𝒳 sup_f_∈𝒴 E(µ, f). We can get

minμ∈𝔐+(K)∥R α,β*μ∥Lp′(ℝn)μ(K)=minμ∈𝔐+(K)supf∈Lp(ℝn)&∥f∥Lp(ℝn)=1∫ℝnf(x)R α,β*μ(x)dxμ(K)≤minμ∈𝔐+(K)supf∈𝒴∫ℝnf(x)R α,β*μ(x)dx∥f∥Lp(ℝn)μ(K)=supf∈𝒴minμ∈𝒳1∥f∥Lp(ℝn){ ∫KRα,βf(x,t)dμ(x,t) }=supf∈𝒴1∥f∥Lp(ℝn)(min(x,t)∈KRα,βf(x,t))minμ∈𝒳μ(K)≤sup0<f∈Lp(ℝn)(min(x,t)∈KRα,βf(x,t))∥f∥Lp(ℝn)≤1inf0≤f∈Lp&minRα,βf=1∥f∥Lp=(C p(α,β)(K))−1/p.

Then for ∥R α,β*μ∥Lp′(ℝn)≤1 , it holds

(C p(α,β)(K))−1/p=minμ∈𝔐+(K)1μ(K)∥R α,β*μ∥Lp′(ℝn).

Recall that

C p(α,β)˜(K)=sup{∥μ∥ 1p: μ∈𝔐+(K) & ∥R α,β*μ∥Lp′(ℝn)≤1}.

For any µ ∈ 𝔐₊(K), take μ1:=∥R α,β*μ∥ Lp′(ℝn)−1μ , which means that ∥R α,β*μ1∥Lp′(ℝn)=1 .

[C p(α,β)˜(K)]1/p≥sup{μ∈𝔐+(K):∥μ∥1∥R α,β*μ∥Lp′(ℝn)}=sup{∥μ1∥1,μ1∈𝔐+(K)},

which implies that

minμ∈𝔐+(K){ ∥R α,β*μ∥Lp′(ℝn)μ(K) }=minμ∈𝔐+(K){ ∥R α,β*μ∥Lp′(ℝn)∥μ∥1 }≥(C p(α,β)˜(K))−1/p.

This gives (C p(α,β)(K))−1/p≥(C p(α,β)˜(K))−1/p . The proof of (i) is completed.

Next, we prove (ii). According to (i), we select a sequence {µ_j} ⊂ 𝔐₊(K) such that

{ limj→∞(μj(K))p=C p(α,β)(K);∥R α,β*μj∥Lp′(ℝn)≤1.

A direct computation implies that

C p(α,β)(K)=sup{ ∥μ∥ 1p: μ∈𝔐+(K) & ∥R α,β*μ∥Lp′(ℝn)=1 }.

Then, using the fact ∥R α,β*μj∥Lp′(ℝn)=1 , we get

| ∫ℝ +n+1Rα,βf(x,t)dμj(x,t) |=| ∫ℝnf(x)R α,β*μj(x)dx |≤∥R α,β*μj∥Lp′(ℝn)∥f∥Lp≤∥f∥Lp.

There exists µ ∈ 𝔐₊(K) such that µ_j weak * convergence to µ. Hence μp(K)=C p(α,β)(K) . Since the Lebesgue space L^p(ℝⁿ) is uniformly convex for 1 < p < ∞, noting that the set

{f∈C 0∞(ℝn): f≥0onℝnandf≥1}

is a convex set. Following the procedure of [3, Theorem 2.3.10], we can prove that there exists a unique function denoted by f_K such that

(4.1) { fK∈Lp(ℝn);C p(α,β)({(x,t)∈K:Rα,βfK<1})=0;∥fK∥ Lp(ℝn)p=C p(α,β)(K).

Then we have

(4.2) (C p(α,β)(K))1/p=μ(K)≤∫ℝ +n+1Rα,βf(x,t)dμ(x,t)≤∫ℝnfK(x)R α,β*μ(x)dx≤∥fK∥Lp(ℝn)∥R α,β*μ∥Lp′(ℝn)=(C p(α,β)(K))1/p∥R α,β*μ∥Lp′(ℝn),

which implies that ∥R α,β*μ∥Lp′(ℝn)≥1 . On the other hand, following the procedure of [3, Proposition 2.3.2, (b)], we can prove that μ↦P α*μ(x) is lower semi-continuous on M+(ℝ +n+1) in the weak* topology, i.e.,

R α,β*μ(x)≤lim infj→∞R α,β*μj(x),

which, together with ∥R α,β*μj∥Lp′(ℝn)≤1 , gives ∥R α,β*μ∥Lp′(ℝn)≤1 . Hence ∥R α,β*μ∥Lp′(ℝn)=1 . Taking μK=(C p(α,β)(K))1/p′μ yields

μK(K)=∫K(C ℝ +n+1α,p(K))1/p′dμ=(C p(α,β)(K))1/p′μ(K)=(C p(α,β)(K))1/p′(C p(α,β)(K))1/p=C p(α,β)(K).

On the other hand,

∫ℝn(Rα,β*μK(x))p′dx=∥Rα,β*μK∥Lp′(ℝn)p′=(Cp(α,β)(K))∥Rα,β*μ∥Lp′(ℝn)p′=Cp(α,β)(K).

This indicates that

(4.3) μK(K)=∫ℝn(Rα,β*μK(x))p′dx=Cp(α,β)(K).

Let f_K be the function mentioned above. Then

μK({(x,t)∈K: PαfK(x,t)≤1})=0.

By Hölder’s inequality, we can get

(4.4) Cp(α,β)(K)=μK(K)≤∫KRα,βfKdμK≤∫ℝnfK(x)Rα,β*μK(x)dx≤∥fK∥Lp(ℝn)∥Rα,β*μK∥Lp′(ℝn)=(Cp(α,β)(K))1/p(Cp(α,β)(K))1/p′=Cp(α,β)(K).

It follows from (4.4) that

Cp(α,β)(K)=∫ℝnfKRα,β*μKdx.

Hence

∫ℝnfKRα,β*μKdx=∫ℝn(Rα,β*μK)p′dx=Cp(α,β)(K).

The above identity implies that

Cℝ+n+1α,p(K)=∫ℝ+n+1Rα,βfKdμK=∫ℝ+n+1Rα,β(Rα,β*μK)p′−1dμK,

which completes the proof of (ii).

Below we investigate some basic properties of Cp(α,β)(⋅) .

Proposition 4.2

Cp(α,β)(∅)=0 ;
If K1⊆K2⊂ℝ+n+1 , then Cp(α,β)(K1)≤Cp(α,β)(K2) ;
For any sequence {Kj}j=1∞ of subsets of ℝ+n+1 ,
Cp(α,β)(∪j=1∞Kj)≤∑j=1∞Cp(α,β)(Kj);
For any K⊂ℝ+n+1 and any x₀ ∈ ℝⁿ, Cp(α,β)(K+(0,x0))=Cp(α,β)(K) .

Proof

By the definition of Cp(α,β) , if K = ∅, then 1_K ≡ 0. Take f = 0. Then 0≤Cp(α,β)(K)≤∥f∥Lp(ℝn)p=0 , i.e., Cp(α,β)(K)=0 .
Let K1⊆K2⊂ℝ+n+1 . For any fixed (x,t)∈ℝ+n+1 , 1_K₁(x, t) ≤ 1_K₂(x, t) due to
{1K1(x,t)=1K2(x,t),(x,t)∈K1;1K1(x,t)=0<1=1K2(x,t),(x,t)∉K1&(x,t)∈K2;1K1(x,t)=1K2(x,t)=0,(x,t)∉K2.
This means that if f ≥ 0 such that R_α,βf ≥ 1_K₂, then R_α,βf ≥ 1_K₂, equivalently,
{∥f∥Lp(ℝn)p: f≥0 & Rα,β(f)≥1K2}⊆{∥f∥Lp(ℝn)p:f≥0&Rα,β(f)≥1K1}.
By the definition of Cp(α,β) , we can get Cp(α,β)(K1)≤Cp(α,β)(K2) .
Let ∊ > 0. Take f_j ≥ 0 such that P_αf_j ≥ 1 on K_j and
∫ℝn|fj(x)|pdx≤Cp(α,β)(Kj)+ɛ/2j.
Let f=supj∈𝕅+fj . For any (x,t)∈∪j=1∞Kj , there exists a j₀ such that (x, t) ∈ K_j₀ and P_αf_j₀(x, t) ≥ 1. Hence P_αf(x, t) ≥ 1 on ∪j=1∞Ej . On the other hand,
∥f∥Lp(ℝn)p=∫ℝn|f(x)|pdx≤∑j=1∞∫ℝn|fj(x)|pdx=∑j=1∞Cp(α,β)(Kj)+ɛ,
which indicates
Cp(α,β)(∪j=1∞Kj)≤∑j=1∞Cp(α,β)(Kj).
Define f_x₀(x) = f(x − x₀). Then ‖f‖_L^p(ℝⁿ) = ‖f_x₀‖_L^p(ℝⁿ). If (x, t) ∈ K + (0, x₀), then (t, x − x₀) ∈ K, and verse visa. Take f ≥ 0 and R_α,βf ≥ 1_K. We have
Rα,βfx0(x,t)=∫ℝnGt(x−y)fx0(y,t)dy=∫ℝnGt(x−y)f(y−x0)dy=∫ℝnGt(x−x0−(y−x0))f(y−x0)dy=Rα,βf(x−x0),
which means that R_α,βf_x₀(x, t) ≥ 1_K+(0,x₀) is equivalent to R_α,βf(x − x₀, t) ≥ 1_K. We have Cp(α,β)(K+(0,x0))=Cp(α,β)(K) .

Proposition 4.3

For t₀ > 0, x₀ ∈ ℝⁿ and r₀ > 0, set the (α, β)-parabolic ball centered at (x₀, t₀) with the radius r₀ as

Br0(α,β)(x0,t0)={(x,t)∈ℝ+n+1, r0α<t−t0<2r0α, |x−x0|<r0β/2}.

If 1 ≤ p < ∞, then
Cp(α,β)(Br0(α,β)(0,0))=r0βnCp(α,β)(B1(α,β)(0,0)), ∀ r0>0.
For t₀ ≥ 0, x₀ ∈ ℝⁿ and r₀ > 0 with t0≲r0α ,
r0nβ≤Cp(α,β)(Br0(α,β)(t0,x0))≤((t0+r0α)nβ/αr0βn)pr0βn.
Specially, for t0≲r0α ,
Cp(α,β)(Br0(α,β)(t0,x0))∼r0βn.

Proof

(i) If f ≥ 0 obeys Rα,βf≥1Br0(α,β)(0,0) , then for (x,t)∈Br0(α)(0,0) , we can see that (x,t)∈Br0(α)(0,0) is equivalent to (x/r0β,t/r0α)∈B1(α,β)(0,0) . Let t/r0α=s and x/r0β=y . If f ≥ 0 and R_α,βf(x, t) ≥ 1, ∀(x,t)∈Br0(α,β)(0,0) , we can get

Rα,βf(x,t)=∫ℝnGt(x−z)f(z)dz=∫ℝn(∫0∞Kα,(t/u)β(x−z)gβ(u)du)f(z)dz=∫ℝn(∫0∞Kα,(r0αs/u)β(x−r0βv)gβ(u)du)f(r0βv)r0βndv.

By the Fourier transform, we can see that, letting r0βξ=η ,

Kα,(r0αs/u)β(x−r0βv)=∫ℝnexp(−r0αβsβuβ|ξ|β)exp(i(x−r0βv)ξ)dξ=∫ℝnexp(i(r0−βx−v)η)exp(−|η|α(s/u)β)r0−βndη=r0−βnKα,(s/u)β(r0−βx−v).

The above identities show that

Rα,βf(x,t)=∫ℝn(∫0∞Kα,(s/u)β(r0−βx−v)gβ(u)du)r0−βf(r0βv)r0βndv=∫ℝnGs(r0−βx−v)fr0(v)dv=Rα,βfr0(r0−βx,s)=Rα,βfr0(s,y),

which indicates that Rα,βf(x,t)≥1 ∀ (x,t)∈Br0(α,β)(0,0) is equivalent to R_α,βf(y, s) ≥ 1, ∀(y,s)∈B1(α,β)(0,0) . Finally, we obtain

Cp(α,β)(Br0(α,β)(0,0))=inf{∥f∥Lp(ℝn)p:f≥0&Rα,βf≥1Br0(α,β)(0,0)}=r0βninf{∥fr0∥Lp(ℝn)p:fr0≥0&Rα,βfr0≥1B1(α,β)(0,0)}≥r0βninf{∥g∥Lp(ℝn)p:g≥0&Rα,βg≥1B1(α,β)(0,0)}=rβnCp(α,β)(B1(α,β)(0,0)).

Conversely, we can also get, changing the order of Cp(α,β)(Br0(α,β)(0,0)) and Cp(α,β)(B1(α,β)(0,0)) ,

r0−βnCp(α,β)(Br0(α,β)(0,0))≤Cp(α,β)(B1(α,β)(0,0)),

which means (i) holds.

(ii) For p ∈ [1, ∞), choose p˜ , q˜ such that

{1≤p<p˜<npn−min{n,α}=∞;1/q˜=nβα(1/p−1/p˜).

If 0 ≤ f ∈ L^p(ℝⁿ) and R_α,βf(x, t) ≥ 1 for (x,t)∈Br0(α,β)(x0,t0)⊂ℝ+n+1 , then

{∫r0α<t−t0<2r0α(∫|x−x0|<r0β/2|Rα,βf(x,t)|p˜dx)q˜/p˜dt}1/q˜ ≥{∫r0α<t−t0<2r0α(∫|x−x0|<r0β/21dx)q˜/p˜dt}1/q˜≥r0βn/p˜+α/q˜.

On the other hand, by Lemma 2.3,

{∫r0α<t−t0<2r0α(∫|x−x0|<r0β/2|Rα,βf(x,t)|p˜dx)q˜/p˜dt}1/q˜≲∥Rα,βf∥Lq˜((t0+r0α,t0+2r0α);Lp˜)≲∥f∥Lp(ℝn).

Since 1/q˜=βn(1/p−1/p˜)/α , the above estimates indicate that r0βn≤∥f∥Lp(ℝn)p .

For the converse, choose f=1{x∈ℝn: |x−x0|<r0β/2} . Then

Rα,βf(x,t)=∫ℝnGt(x−y)f(y)dy=∫{y∈ℝn: |y−x0|<r0β/2}Gt(x−y)f(y)dy.

If (x,t)∈Br0(α,β)(x0,t0) , then |x−x0|<r0β/2 . Since t0+r0α<t<t0+2r0α ,

tβ(tβ/α+|x−y|)n+α≥1tβn/α≥1(t0+r0α)nβ/α,

which gives

Rα,β(f)(x,t)≥∫{y∈ℝn: |y−x0|<r0β/2}tβ(tβ/α+|x−y|)n+αdy≥r0nβ(t0+r0α)nβ/α,

equivalently, Rα,β((t0+r0α)nβ/αr0−βnf)≥1 . Under the assumption t0≤r0α , we obtain

Cp(α,β)(Br0(α,β)(x0,t0))≲‖((t0+r0α)nβ/αr0−βn)f(x)‖Lp(ℝn)p=((t0+r0α)nβ/αr0−βn)pr0βn.

Specially, for t0≤r0α , Cp(α,β)(Br0(α,β)(x0,t0))∼r0βn . This completes the proof of Proposition 4.3.

We investigate the capacitary strong and weak type inequalities. As some preliminaries, we prove several further properties of the capacity Cp(α,β)(⋅) . At first, the capacity Cp(α,β)(⋅) is an outer capacity, i.e.,

Proposition 4.4

For any E⊂ℝ+n+1 ,

Cp(α,β)(E)=inf{Cp(α,β)(O):O⊃E,Oopen}.

Proof

Without loss of generality, we assume that Cp(α,β)(E)<∞ . By (ii) of Proposition 4.2,

Cp(α,β)(E)≤inf{Cp(α,β)(O):O⊃E,Oopen}.

For ∊ ∈ (0, 1), there exists a measurable, nonnegative function f such that R_α,βf ≥ 1 on E and

∫ℝn|f(x)|pdx≤Cp(α,β)(E)+ɛ.

Since R_α,βf is lower semi-continuous, then the set Oɛ:={(x,t)∈ℝ+n+1:Rα,βf(x,t)>1−ɛ} is an open set. On the other hand, E ⊂ O_∊. This implies that

Cp(α,β)(Oɛ)≤1(1−ɛ)p∫ℝn|f(x)|pdx<1(1−ɛ)p(Cp(α,β)(E)+ɛ).

The arbitrariness of ∊ indicates that Cp(α,β)(E)≥inf{Cp(α,β)(O):O⊃E,Oopen} .

An immediate corollary of Proposition 4.4 is the following result.

Corollary 4.5

If {Kj}j=1∞ is a decreasing sequence of compact sets, then

Cp(α,β)(∩j=1∞Kj)=limj→∞Cp(α,β)(Kj).

Proposition 4.6

Let 1 < p < ∞. If {Ej}j=1∞ is an increasing sequence of arbitrary subsets of ℝⁿ, then Cp(α,β)(∪j=1∞Ej)=limj→∞Cp(α,β)(Ej) .

Proof

Since {Ej}j=1∞ is increasing, then Cp(α,β)(∪j=1∞Ej)≥limj→∞Cp(α,β)(Ej) .

Conversely, without loss generality, we assume that limj→∞Cp(α,β)(Ej) is finite. For each j, let f_{E_j} be the unique function such that f_{E_j} ≥ 1 on E_j and ∥fEj∥Lpp=Cp(α,β)(Ej) . Then for i < j, it holds that R_α,βf_{E_j} ≥ 1 on E_i and further, R_α,β((f_{E_i} + f_{E_j})/2) ≥ 1 on E_i, which means that

∫ℝn((fEi+fEj)/2)pdx≥Cp(α,β)(Ei).

It follows from [3, Corollary 1.3.3] that the sequence {fEj}j=1∞ converges strongly to a function f satisfying

∥f∥Lpp=limj→∞Cp(α,β)(Ej).

Similar to [3, Proposition 2.3.12], it can be deduced that R_α,βf ≥ 1 on ∪j=1∞Ej , except possibly on a countable union of sets with Cp(α,β)(⋅) zero. Hence

limj→∞Cp(α,β)(Ej)≥∫ℝn|f(x)|pdx≥Cp(α,β)(∪j=1∞Ej).

As a corollary of Proposition 4.6, we can get

Corollary 4.7

Let O be an open subset of ℝ+n+1 . Then

Cp(α,β)(O)=sup{Cp(α,β)(K):compactK⊂O}.

The strong type inequality corresponding to Cp(α,β)(⋅) is as follows.

Lemma 4.8

Let p ∈ (1, ∞). Then

∫0∞Cp(α,β)({(x,t)∈ℝ+n+1: Rα,βf(x,t)≥λ})dλp≲∥f∥Lp(ℝn)p, ∀f∈L+p(ℝn).

Here and henceforth, the symbol L+p(ℝn) stands for the class of all nonnegative functions in L^p(ℝⁿ) and dλ^p = pλ^p−1dλ.

Proof

Since C0∞(ℝn)↪Lp(ℝn) , we are about to verify this inequality for any nonnegative C0∞(ℝn) -function. Assume that f∈C0∞(ℝn) . For r > 0, let

Ej:={(x,t)∈Br(0,0)¯:Rα,βf(x,t)≥2j}.

Note that E_j depends on r. Since f∈C0∞(ℝn) , R_α,βf(x, t) is continuous. Thus, E_j is the intersection of Br(0,0)¯ and a close set which is equivalent to a compact set in ℝⁿ.

Let µ_j stand for the measure corresponding to E_j such that

μj(Ej)=∫ℝn(Rα,β*μj(x))p′dx=∫ℝ+n+1Rα,β(Rα,β*μj)p′−1dμj=Cp(α,β)(Ej).

Let S:=∑j=−∞∞2jpμj(Ej) and T:=∥∑j=−∞∞2j(p−1)(Rα,β*μj)∥Lp′p′ . Because f∈C0∞(ℝn) , there exists a positive integer j₀ such that E_j are empty sets for j > j₀, i.e., μj(Ej)=Cp(α,β)(Ej)=0 , j > j₀. Hence ∑j=1∞2jpμj(Ej)<∞ . On the other hand,

∑j=∞−12jpμj(Ej)=∑j=∞−12jp∑j=∞−12jpCp(α,β)(Ej)≤∑j=∞−12jp∑j=∞−12jpCp(α,β)(Br(0,0))≲rn∑j=∞−12jp∑j=∞−12jp,

which gives S < ∞. It follows from Hölder’s inequality that

(4.5) S≤∑j=−∞∞2j(p−1)∫ℝ+n+1Rα,βfdμj≤∫ℝnf(∑j=−∞∞2j(p−1)(Rα,β*μj))dx≤T1/p′∥f∥Lp.

Below we prove that

(4.6) T≲S,

The proof of (4.6) is divided into two cases.

Case 1: 2 ≤ p < ∞. For k = 0, ±1, ±2, . . . , let

{σk(x)=∑j=k∞2j(p−1)Rα,β*μj(x);σ(x)=∑j=−∞∞2j(p−1)Rα,β*μj(x).

Since f∈C0∞(ℝn) , the sets E_j are empty for sufficiently large j, i.e., there exists a positive integer j₀ such that Cp(α,β)(Ej)=0 , j > j₀. Then by (ii) of Proposition 4.1,

∥σk∥Lp′=∥∑j=k∞2j(p−1)Rα,β*μj∥Lp′≤∑j=k∞2j(p−1)‖Rα,β*μj‖Lp′≃∑j=kj02j(p−1)(Cp(α,β)(Ej))1/p′<∞,

which implies that for any k, σ_k ∈ L^p′ (ℝⁿ). For x ∈ ℝⁿ, the sequence {σ_k(x)} is increasing as k → −∞ since Rα,β*μj(x)≥0 . If there exists an upper bound for {σ_k(x)}, we know that σ_k(x) converges to the series ∑j=−∞∞2j(p−1)Rα,β*μj(x) . If the sequence {σ_k} is unbounded, then σ_k tends to ∞ as k → −∞. Without loss of generality, formally, we write limk→−∞σk(x)=σ(x) . Then by the mean value theorem, noticing that σ_k(x) ≥ σ_k₊₁(x), we have

(4.7) σ(x)p′=(∑j=−∞∞2j(p−1)Rα,β*μj(x))p′=p′∑k=−∞∞(σk(x))p′−1=limn→−∞σn(x)p′=limn→−∞∑k=n∞(σk(x)p′−σk+1(x)p′)≤p′(σk(x))p′−1(σk(x)−σk+1(x))=p′∑k=−∞∞(σk(x))p′−12k(p−1)Rα,β*μk(x).

We use Cauchy-Schwartz’s inequality to obtain

T=p′∫ℝn∑k=−∞∞σkp′−1(x)2k(p−1)Rα,β*μk(x)dx≲p′∫ℝn(∑k=−∞∞(σk(x))2k1(p−1)1p′−1(Rα,β*μk)p′−1)p′−1×(∑k=−∞∞2k(p−1)p(p−2)(p−1)212−p′(Rα,β*μk(x))p′)2−p′dx=p′∫ℝn(∑k=−∞∞σk(x)2k(Rα,β*μk)p′−1)p′−1(∑k=−∞∞2kp(Rα,β*μk(x))p′)2−p′dx,

which, together with Hölder’s inequality, indicates that T≲p′T12−p′T2p′−1 , where

{T1:=∫ℝn(∑k=−∞∞2kp(Rα,β*μk(x))p′)dx;T2:=∫ℝn(∑k=−∞∞σk(x)2k(Rα,β*μk)p′−1)dx.

For T₁, we have

T1=∑k=−∞∞2kp∫ℝn(Rα,β*μk(x))p′dx=∑k=−∞∞2kpCp(α,β)(Ek)≲∑k=−∞∞∫2k−12kCp(α,β)({(x,t)∈Br(0,0)¯:Rα,βf(x,t)≥2k})dλp≲∑k=−∞∞∫2k−12kCp(α,β)({(x,t)∈Br(0,0)¯:Rα,βf(x,t)≥λ})dλp≲∫0∞Cp(α,β)({(x,t)∈Br(0,0)¯: Rα,βf(x,t)≥λ})dλp≲S.

For T₂, we can get

T2=∑k=−∞∞2k∫ℝn(∑j≥k∞2j(p−1)Rα,β*μj(x))(Rα,β*μk)p′−1dx=∑k=−∞∞∑j≥k∞2k+j(p−1)∫ℝnRα,β*μj(x)(Rα,β*μk)p′−1dx=∑k=−∞∞∑j≥k∞2k+j(p−1)∫ℝ+n+1Rα,β(Rα,β*μk)p′−1dμj(x).

Note that E_k is compact subsets. Let f_{E_k} be the function satisfying (4.1). Suppose that R_α,βf_{E_k}(x₀, t₀) > 1, by the lower-semicontinuity, R_α,βf_{E_j} ≥ 1 + δ > 1 on some neighborhood U of (x₀, t₀). On the other hand, denote by Ω_{E_k} the set {(x, t) ∈ E_k : R_α,βf_{E_k} < 1} and let F be any compact subset of Ω_{E_k}. For any 0 ≤ f ∈ L^p(ℝⁿ) such that R_α,βf ≥ 1 on F

μk(F)≤∫ℝ+n+1Rα,βfdμk=∫ℝnfRα,β*μkdx≤∥Rα,β*μk∥Lp′∥f∥Lp,

which, together with (4.3), gives

μk(F)≤∥Rα,β*μk∥Lp′Cp(α,β)(F)≤∥Rα,β*μk∥Lp′Cp(α,β)(ΩEk)=0.

Hence, µ_k(Ω_{E_k}) = 0, i.e., R_α,βf_{E_k} ≥ 1 on E_k for µ_k a.e. We can deduce that

Cp(α,β)(Ek)=∫ℝ+n+1Rα,β(Rα,β*μk)p′−1dμk=∫ℝ+n+1Rα,βfEkdμk≥(1+δ)μk(U)+μk(Ek\U)≥δμk(U)+μk(Ek),

which gives µ_k(U) = 0, i.e., (x₀, t₀) ∉ supp µ_k. Equivalently, for all (x, t) ∈ supp µ_k,

Rα,β(Rα,β*μk)p′−1=Rα,βfEk≤1.

Since for j ≥ k, E_j ⊂ E_k and we obtain

T2≲∑k=−∞∞∑j≥k∞2k+j(p−1)Cp(α,β)(Ej)=∑j=−∞∞2j(p−1)Cp(α,β)(Ej)∑k=−∞j2k≲∑j=−∞∞2jpCp(α,β)(Ej)≲S.

The estimates for T₁ and T₂ yields (4.6). It can be deduced from (4.5) & (4.6) that

∥∑j=−∞∞2j(p−1)(Rα,β*μj)∥Lp′p′≲∥f∥Lpp,

i.e., σ ∈ L^p′(ℝⁿ) and the series ∑j=−∞∞2j(p−1)Rα,β*μj(x)<∞ a.e. x ∈ ℝⁿ.

Case 2: 1 < p < 2. For k = 0, ±1, ±2, . . . , let

{σk(x)=∑j=−∞k2j(p−1)Rα,β*μj(x);σ(x)=∑j=−∞∞2j(p−1)Rα,β*μj(x).

Similarly, because σ_k(x) ≥ 0 for x ∈ ℝⁿ, we can also write limk→∞σ(x)=σ(x) formally. Below, similar to Case 1, we will prove (4.6) and σ ∈ L^p′(ℝⁿ). Following the procedure of (4.7), we can deduce that

(σ(x))p′=(∑j=−∞∞2j(p−1)Rα,β*μj(x))p′≤p′∑k=−∞∞(σk(x))p′−12k(p−1)Rα,β*μk(x).

We get

T=p′∑k=−∞∞2k(p−1)∫ℝnPα*μk(x)(∑j=−∞k2j(p−1)Rα,β*μj(x))p′−1dx=p′∑k=−∞∞2k(p−1)∫ℝn(∑j=−∞k2j(p−1)(Rα,β*μk(x))1/(p′−1)Rα,β*μj(x))p′−1dx=p′∑k=−∞∞2k(p−1)‖∑j=−∞k2j(p−1)(Rα,β*μk(x))1/(p′−1)Rα,β*μj(x)‖Lp′−1p′−1≲p′∑k=−∞∞2k(p−1){∑j=−∞k2j(p−1)‖(Rα,β*μk(x))1/(p′−1)Rα,β*μj(x)‖Lp′−1}p′−1.

Also, for j ≤ k, if x ∈ E_k then x ∈ E_j. Similarly, we can deduce that Rα,β(Rα,β*μj(x))p′−1(x)≤1 , x ∈ E_k. This indicates that

∫ℝn(Rα,β*μk(x))(Rα,β*μj(x))p′−1dx=∫ℝ+n+1Rα,β(Rα,β*μj(x))p′−1dμk≤μk(Ek)=Cp(α,β)(Ek).

We obtain

T≲∑k=−∞∞2k(p−1){∑j=−∞k2j(p−1)(∫ℝnRα,β*μk(x)(Rα,β*μj(x))p′−1dx)1/(p′−1)}p′−1≲∑k=−∞∞2k(p−1)Cp(α,β)(Ek)(∑j=−∞k2j(p−1))p′−1≲∑k=−∞∞2kpCp(α,β)(Ek)≲S.

which gives (4.6) for 1 < p < 2.

It follows from (4.5) & (4.6) that S ≲ ‖f‖_L^p T^1/p′ ≲ ‖f‖ _L^pS^1/p′. Then we can get S≲∥f∥Lpp and consequently, T < ∞. Then σ ∈ L^p′(ℝⁿ) for 1 < p < 2. By the fact that f∈C0∞(ℝn) , there exists an integer j₀ such that

∥∑j=0∞2j(p−1)Rα,β*μj∥Lp′≤∑j=0∞2j(p−1)‖Rα,β*μj‖Lp′≃∑j=0j02j(p−1)(Cp(α,β)(Ej))1/p′<∞.

This indicates that ∑j=−∞−12j(p−1)Rα,β*μj∈Lp′(ℝn) . For σ_k, if k ≤ −1, because Rα,β*μj(x)≥0 , then

∥σk∥Lp′≤∥∑j=−∞−12j(p−1)Rα,β*μj∥Lp′

and σ_k ∈ L^p′(ℝⁿ). If k ≥ 0, we can also use the identity

σk=∑j=−∞−12j(p−1)Rα,β*μj+∑j=0j02j(p−1)Rα,β*μj,

to get σ_k ∈ L^p(ℝⁿ). Further, we obtain that

∫0∞Cp(α,β)({(x,t)∈Br(0,0)¯: Rα,βf(x,t)≥λ})dλp =∑j=−∞∞∫2j2j+1Cp(α,β)({(x,t)∈Br(0,0)¯: Rα,βf(x,t)≥λ})dλp ≲∑j=−∞∞Cp(α,β)(Ej)∫2j2j+1dλp ≲∑j=−∞∞2pjμj(Ej)≲∥f∥Lpp.

Hence we can deduce from Proposition 4.6 and Fauto’s lemma that

∫0∞Cp(α,β)({(x,t)∈ℝ+n+1:Rα,βf(x,t)>λ})dλp =∫0∞Cp(α,β)(∪r=1∞{(x,t)∈Br(0,0):Rα,βf(x,t)>λ})dλp =∫0∞limr→∞Cp(α,β)({(x,t)∈Br(0,0):Rα,βf(x,t)>λ})dλp ≤lim_r→∞∫0∞Cp(α,β)({(x,t)∈Br(0,0)¯:Rα,βf(x,t)≥λ})dλp≲∥f∥Lpp.

By the above capacitary strong type inequality, it is easy to get the following capacitary weak type inequality

λpCp(α,β)({(x,t)∈ℝ+n+1: Rα,βf(x)≥λ})≤∥f∥Lp(ℝn)p, ∀f∈L+p(ℝn).

5 Lq(ℝ+n+1,μ) -extension of L^p(ℝⁿ)

Let 1 < p ≤ q < ∞ and μ∈𝒨+(ℝ+n+1) . For λ > 0, define

κ(μ;λ):=inf{Cp(α,β)(K):compactK⊂ℝ+n+1,μ(K)≥λ}.

Theorem 5.1

Let 1 < p ≤ q < ∞ and μ∈𝒨+(ℝ+n+1) .

The extension Rα,β:Lp(ℝn)→Lq(ℝ+n+1) is bounded if and only if
supλ∈(0,∞)λp/qκ(μ;λ)<∞;
If 1 < p < q < ∞, then λ^p/q ≲ κ(µ; λ) ∀λ ∈ (0, ∞) can be replaced by μ(Br(α,β)(x0,t0))≲rβqn/p∀(r,t0,x0)∈ℝ+×ℝ+×ℝn with t₀ ≲ r^α.

Proof

(i). Suppose that R_α,β is bounded from L^p(ℝⁿ) to Lq(ℝ+n+1,μ) . For a compact set K⊂ℝ+n+1 , denote by µ |_K the restriction of µ to the set K. Then

∫ℝnf(x)Rα,β*μ|Kdx=∬ℝ+n+1Rα,βf(x,t)dμ|K(x,t)≤{∬ℝ+n+1(Rα,βf(x,t))qdμ|K(x,t)}1/q{∬ℝ+n+1dμ|K(x,t)}1−1/q=∥Rα,βf∥Lq(ℝ+n+1,μ)(μ(K))1−1/q≲∥f∥Lp(ℝn)(μ(K))1−1/q,

which implies that ∥Rα,β*μ|K∥Lp′(ℝn)≲(μ(K))1/q′ . Letting

Eλ(f):={(x,t)∈ℝ+n+1: |Rα,βf(x,t)|>λ},

we have

which implies that λ(µ(E_λ(f)))^1/q ≲ ‖f‖_L^p(ℝⁿ) and, hence, supλ∈(0,∞)λqμ(Eλ(f))≲∥f∥Lp(ℝn)q . Choose a function f ∈ L^p(ℝⁿ) such that R_α,βf ≥ 1 on a given compact set K⊂ℝ+n+1 , that is, K ⊂ E₁(f). It follows from the above estimate that

(μ(K))1/q≤(μ(E1(f)))1/q≲∥f∥Lp(ℝn)

and

(μ(K))p/q≲inf{∥f∥Lp(ℝn)p: f≥0 & Rα,βf≥1}≤Cp(α,β)(K).

If K is compact and µ(K) ≥ λ, then

λ1/q≤(μ(K))1/q≤(Cp(α,β)(K))1/p.

This means that λp/q≤Cp(α,β)(K) , equivalently,

λp/q≤inf{Cp(α,β)(K): Kiscompactandμ(K)≥λ},

that is, λ^p/q ≲ κ(µ, λ).

Conversely, we assume that sup_λ_∈_(0,∞) λ^p/q/κ(µ, λ) < ∞. We know that for any compact set K, (μ(K))1/q≲(Cp(α,β)(K))1/p . We have proved that

(5.1) ∫0∞Cp(α,β)({(x,t)∈ℝ+n+1:Rα,βf(x,t)≥λ})dλp≲∥f∥Lp(ℝn)p.

For any τ > 0, we can get

(5.2) τpCp(α,β)(Eτ)=τpCp(α,β)({(x,t)∈ℝ+n+1: Rα,βf(x,t)≥τ})=∫0τCp(α,β)({(x,t)∈ℝ+n+1: Rα,βf(x,t)≥τ})dλp≲∫0τCp(α,β)({(x,t)∈ℝ+n+1: Rα,βf(x,t)≥λ})dλp≲∫0∞Cp(α,β)({(x,t)∈ℝ+n+1: Rα,βf(x,t)≥λ})dλp≲∥f∥Lp(ℝn)p.

We can use (5.1) & (5.2) to obtain

∬ℝ+n+1|Rα,βf(x,t)|qdμ(x,t)=μ(Eλ)∫0∞dλq≤∫0∞(Cp(α,β)(Eλ))q/pdλq=q∫0∞(Cp(α,β)(Eλ))q/p−1(Cp(α,β)(Eλ))λq−1dλ≲∫0∞(λ−p∥f∥Lp(ℝn)p)q/p−1(Cp(α,β)(Eλ))λq−1dλ≲∥f∥Lp(ℝn)q−p∫0∞Cp(α,β)(Eλ)dλp≲∥f∥Lp(ℝn)q,

which completes the proof of (i).

(ii). If λ^p/q ≤ κ(µ, λ), then for any set K⊂ℝ+n+1 , (μ(K))1/q≤(Cp(α,β)(K))1/p . Let K=Br(α,β)(t0,x0) . Then

(μ(Br(α,β)(x0,t0)))1/q≤(Cp(α,β)(Br(α,β)(x0,t0)))1/p.

By Proposition 4.3,

rnβ≲Cp(α,β)(Br(α,β)(x0,t0))≲((t0+rα)nβ/αr0−βn)prβn.

Then for t₀ ≤ r^α, Cp(α,β)(Br(α,β)(x0,t0))≲rnβ and (μ(Br(α,β)(x0,t0)))1/q≲rnβ/p .

For the reverse implication, if (x,t)∈Br(α,β)(x0,t0) , then |x − x₀| ≤ r^β/2 and r^α < t − t₀ < 2r^α. We get t > t₀ + r^α > r^α and |x − x₀| < r^β/2 ≲ t^β/α, and hence,

Gt(x−y)≥tβ(tβ/α+|x−x0|)n+α=1tnβ/α1(1+|x−x0|/tβ/α)n+α≥1tnβ/α≥1rnβ.

This means that if (x,t)∈Br(α,β)(x0,t0) then r ≥ (G_t(x − x₀))^−1/nβ. We obtain

Minkowski’s inequality gives

∥Rα,β*μ|K∥Lp′(ℝn)≲∫0∞∥μ|K(Br(α,β)(⋅,t0))∥Lp′(ℝn)drrnβ+1.

On the other hand,

‖μ|K(Br(α,β)(⋅,t0))‖Lp′(ℝn)p′=∫ℝn(μ|K(Br(α,β)(x,t0)))p′dx≲(μ(K))p′−1∫ℝnμ|K(Br(α,β)(x,t0))dx≲(μ(K))p′rβn.

Hence,

∫δ∞∥μ|K(Br(α,β)(x0,t0))∥Lp′(ℝn)drrnβ+1≲∫δ∞μ(K)rnβ/p′−nβ−1dr≲μ(K)δ−nβ/p.

Meanwhile, using μ(Br(α,β)(x0,t0))≲rnβq/p , we have

‖μ|K(Br(α,β)(x0,t0))‖Lp′(ℝn)p′=∫ℝn(μ|K(Br(α,β)(x0,t0)))p′dx0≲rnβq(p′−1)/p∫ℝnμ|K(Br(α,β)(x0,t0))dx0≲rnβq(p′−1)/pμ(K)rnβ≲rnβ(1+q/p(p−1))μ(K),

which gives

∫0δ‖μ|K(Br(α,β)(x0,t0))‖Lp′(ℝn)drrnβ+1≲(μ(K))1/p′∫0δrnβ(1+q/p(p−1))/p′drrnβ+1≲(μ(K))1/p′∫0δrnβ/p′+nβq/p2−nβ−1dr≲(μ(K))1/p′δnβ(q−p)/p2.

Finally,

∥Rα,β*μ|K∥Lp′(ℝn)≲μ(K)δ−nβ/p+(μ(K))1/p′δnβ(q−p)/p2.

Take δ such that (µ(K))δ^−nβ/p = (µ(K))^1/p′δ^{nβ(q−p)/p²}, that is, δ = (µ(K))^p/nβq. For such a δ,

∥Rα,β*μ|K∥Lp′(ℝn)≲μ(K)((μ(K))p/nβq)−nβ/p≲(μ(K))1/q′.

Let f∈C0∞(ℝn) and E_λ = {x : |R_α,β * f(x, t)| ≥ λ}. Then the set E_λ is compact. It follows from the above estimate that

(5.3) λμ(Eλ)≤∫ℝn|Rα,βf(x,t)|dμ|Eλ≤∥f∥Lp∥Rα,β*μ|Eλ∥Lp′≲∥f∥Lp(μ(Eλ))1/q′.

For an arbitrary f ∈ L^p(ℝⁿ), via approximating f by a sequence from C0∞(ℝn) in the L^p-norm, we can prove that (5.3) holds for f. For f ∈ L^p(ℝⁿ) such that R_α,βf ≥ 1 on K. By (5.3), it holds (µ(K))^1/q ≲ ‖f‖_L^p, which gives (μ(K))1/q≤(Cp(α,β)(K))1/p . Recall that µ(K) ≥ λ. Then taking the infimum over the compact sets K such that µ(K) ≥ λ, we get λp/q≲Cp(α,β)(K) , i.e.,

λp/q≲inf{Cp(α,β)(K): compactK⊂ℝ+n+1, μ(K)≥λ}=cα,p(μ;λ).

This completes the proof of Theorem 5.1.

Theorem 5.2

Let 1 < q < p < ∞ and μ∈𝒨+(ℝ+n+1) . The extension

Rα,β:Lp(ℝn)→Lq(ℝ+n+1,μ)

is bounded if and only if

∫0∞(λp/qκ(μ;λ))q/(p−q)dλλ<∞.

Proof

We first assume that Rα,β:Lp(ℝn)→Lq(ℝ+n+1,μ) is bounded. Then

(∬ℝ+n+1|Rα,βf(x,t)|qdμ)1/q≲∥f∥Lp(ℝn).

We can get

supλ>0λ(μ(Eλ(f)))1/q=supλ>0{λqμ(Eλ(f))}1/q≲supλ>0{λq∫Eλ(f)1dμ(x,t)}1/q≲supλ>0{∫Eλ(f)|Rα,βf(x,t)|qdμ(x,t)}1/q≲{∫ℝ+n+1|Rα,βf(x,t)|qdμ(x,t)}1/q≲∥f∥Lp(ℝn).

By the definition of κ(µ, 2^j), for any j ∈ 𝕑, there exists K_j such that µ(K_j) > 2^j and Cp(α,β)(Kj)≤2κ(μ,2j) . On the other hand, since

Cp(α,β)(Kj)=inf{∥f∥Lp(ℝn)p: f≥0,Rα,βf≥1Kj}.

There exists a function f_j ∈ L^p(ℝⁿ) such that R_α,βf_j ≥ 1_{K_j} and ∥fj∥Lp(ℝn)p≤2Cp(α,β)(Kj) .

For the integers i, k with i < k, let

fi,k:=supi≤j≤k(2jκ(μ;2j))1/(p−q)fj.

Then

∥fi,k∥Lp(ℝn)p≤∑j=ik(2jκ(μ;2j))p/(p−q)∥fj∥Lp(ℝn)p≲∑j=ik(2jκ(μ;2j))p/(p−q)Cp(α,β)(Kj)≲∑j=ik(2jκ(μ;2j))p/(p−q)κ(μ;2j).

Note that for i ≤ j ≤ k, if (x, t) ∈ K_j, then

We can see from the above estimate that

Kj⊂{(x,t)∈ℝ+n+1:Rα,βfi,k(x,t)>(2jκ(μ;2j))1/(p−q)}.

This means that

2j<μ(Kj)<μ(E(2j/κ(μ;2j))1/(p−q)(fi,k)).

We can obtain

∥fi,k∥Lp(ℝn)p≥∬ℝ+n+1|Rα,β(fi,k)(x,t)|qdμ(x,t)≃∫0∞(inf{λ: μ(Eλ(fi,k))≤s})qds≥∑j=ik2j(inf{λ: μ(Eλ(fi,k))≤2j})q≥∑j=ik2j(2jκ(μ;2j))q/(p−q)≥(∑j=ik(2j/κ(μ;2j))q/(p−q)2j(∑j=ik(2j/κ(μ;2j))p/(p−q)κ(μ;2j))q/p)∥fi,k∥Lp(ℝn)q≃(∑j=ik2jp/(p−q)(κ(μ;2j))q/(p−q))1−q/p∥fi,k∥Lp(ℝn)q.

This implies

∫0∞(λp/q/κ(μ;λ))q/(p−q)λ−1 dλ≲∑j=−∞∞2jp/(p−q)(κ(μ;2j))q/(p−q)≲1.

Conversely, let

Ip,q(μ)=∫0∞(λp/qκ(μ;λ))q/(p−q)dλλ<∞.

Now for each integer j = 0, ±1, ±2, ..., and f∈C0(ℝn) , let

Sp,q(μ;f)=∑j=−∞∞{μ(E2j(f))−μ(E2j+1(f))}p/(p−q)(Cp(α,β)(E2j(f)))q/(p−q).

Using integration-by-part, Hölder’s inequality and Lemma 4.8, we obtain

∫ℝ+n+1|Rα,βf|qdμ =−∫0∞λq dμ(Eλ(f)) ≲∑j=−∞∞{μ(E2j(f))−μ(E2j+1(f))}2jq ≲(Sp,q(μ;f))1−q/p(∑j=−∞∞2jpCp(α,β)(E2j(f)))q/p ≲(Sp,q(μ;f))1−q/p(∫0∞Cp(α,β)({(x,t)∈ℝ+n+1: |Rα,βf(x,t)|>λ}) dλp)q/p ≲(Sp,q(μ;f))1−q/p∥f∥Lp(ℝn)q.

Note also that

(Sp,q(μ;f))1−q/p =(∑j=−∞∞{μ(E2j(f))−μ(E2j+1(f))}p/(p−q)(Cp(α,β)(E2j(f)))q/(p−q))1−q/p =(∑j=−∞∞{μ(E2j(f))−μ(E2j+1(f))}p/(p−q)(κ(μ;μ(E2j(f))))q/(p−q))1−q/p =(∑j=−∞∞(ν(E2j(f)))p/(p−q)−(μ(E2j+1(f)))p/(p−q)(κ(μ;μ(E2j(f))))q/(p−q))1−q/p ≲(∫0∞dsp/(p−q)(κ(μ;s))q/(p−q))1−q/p ≃(Ip,q(μ))1−q/p.

Therefore,

(∬ℝ+n+1|Rα,βf(x,t)|q dμ)1/q≲(Ip,q(μ))(p−q)/pq∥f∥Lp(ℝn).

Acknowledgements

This work was in part supported by National Natural Science Foundation of China (Nos. 11871293, 12071272) and Shandong Natural Science Foundation of China (No. ZR2020MA004).

Conflict of interest statement:
Authors state no con˛ict of interest.
Data availability statements:
Deposition of research data in controlled access repositories.

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Received: 2021-04-03

Accepted: 2022-01-02

Published Online: 2022-02-14

This work is licensed under the Creative Commons Attribution 4.0 International License.

Application of Capacities to Space-Time Fractional Dissipative Equations II: Carleson Measure Characterization for Lq(ℝ+n+1,μ) −Extension

Abstract

1 Introduction

Definition 1.1

Definition 1.2

Definition 1.3

2 Preliminaries on fractional heat kernels and Sobolev capacities

Remark 2.1

Proposition 2.2

Lemma 2.3

Proof

Proposition 2.4

Lemma 2.5

Proof

Lemma 2.6

Proof

Lemma 2.7

Lemma 2.8

3 Lq(ℝ+n+1,μ) −extension of Ẇγ,p(ℝn)

3.1 Lq(ℝ +n+1,μ) −extension of Ẇγ,p(ℝn) when p ≤ q

Theorem 3.1

Proof

Theorem 3.2

Proof

Corollary 3.3

Theorem 3.4

3.2 Lq(ℝ +n+1,μ) −extension of Ẇγ,p(ℝn) when p > q

Theorem 3.5

Proof

Theorem 3.6

Proposition 3.7

Proof

4 Lp-capacity for space-time fractional heat equations

Proposition 4.1

Proof

Proposition 4.2

Proof

Proposition 4.3

Proof

Proposition 4.4

Proof

Corollary 4.5

Proposition 4.6

Proof

Corollary 4.7

Lemma 4.8

Proof

5 Lq(ℝ+n+1,μ) -extension of Lp(ℝn)

Theorem 5.1

Proof

Theorem 5.2

Proof

Acknowledgements

References

Journal and Issue

Articles in the same Issue

3 Lq(ℝ+n+1,μ) −extension of Ẇ^γ,p(ℝⁿ)

3.1 Lq(ℝ +n+1,μ) −extension of Ẇ^γ,p(ℝⁿ) when p ≤ q

3.2 Lq(ℝ +n+1,μ) −extension of Ẇ^γ,p(ℝⁿ) when p > q

4 L^p-capacity for space-time fractional heat equations

5 Lq(ℝ+n+1,μ) -extension of L^p(ℝⁿ)