Let G be a connected and finite graph with the set of vertices V and the set of edges E. Let \(M_{G}\) be the Hardy–Littlewood maximal function defined on graph G and \(M_{\upalpha ,G}\)\((0\le \upalpha <1)\) be its fractional version. In this paper, the regularity problems related to \(M_{G}\) and \(M_{\upalpha ,G}\) will be studied. We show that \(M_{G}:\mathrm{BV}_p (G)\rightarrow \mathrm{BV}_p(G)\)

In this paper, the \(L^2 \times L^{\infty } \rightarrow L^2\) and \(L^2 \times L^2 \rightarrow L^1\) estimates of bilinear Fourier multiplier operators are discussed under weak smoothness conditions on multipliers. As an application, we prove the \(L^2 \times BMO \rightarrow L^2\) and \(L^2 \times L^2 \rightarrow H^1\) boundedness of bilinear operators with multipliers of limited smoothness satisfying

We give two-weighted norm estimates for higher order commutator of classical operators such as singular integral and fractional type operators, between weighted \(L^p\) and certain spaces that include Lipschitz, BMO and Morrey spaces. We also give the optimal parameters involved with these results, where the optimality is understood in the sense that the parameters defining the corresponding spaces

The objective of this work is an existence proof for variational solutions u to parabolic minimizing problems. Here, the functions being considered are defined on a metric measure space \(({\mathcal {X}}, d, \mu )\). For such parabolic minimizers that coincide with Cauchy-Dirichlet data \(\eta \) on the parabolic boundary of a space-time-cylinder \(\varOmega \times (0, T)\) with an open subset \(\varOmega

We study the nuclearity of the Gelfand–Shilov spaces \({\mathcal {S}}^{({\mathfrak {M}})}_{({\mathscr {W}})}\) and \({\mathcal {S}}^{\{{\mathfrak {M}}\}}_{\{{\mathscr {W}}\}}\), defined via a weight (multi-)sequence system \({\mathfrak {M}}\) and a weight function system \({\mathscr {W}}\). We obtain characterizations of nuclearity for these function spaces that are counterparts of those for Köthe

We consider the dynamics of transcendental self-maps of the punctured plane, \(\mathbb {C}^*=\mathbb {C}{\setminus } \{0\}\). We prove that the escaping set \(I(f)\) is either connected, or has infinitely many components. We also show that \(I(f)\cup \{0,\infty \}\) is either connected, or has exactly two components, one containing 0 and the other \(\infty \). This gives a trichotomy regarding the

We give an algebraic and self-contained proof of the existence of the so-called Noetherian operators for primary submodules over general classes of Noetherian commutative rings. The existence of Noetherian operators accounts to provide an equivalent description of primary submodules in terms of differential operators. As a consequence, we introduce a new notion of differential powers which coincides

For a monomial ideal I, we consider the ith homological shift ideal of I, denoted by \({\text {HS}}_i(I)\), that is, the ideal generated by the ith multigraded shifts of I. Some algebraic properties of this ideal are studied. It is shown that for any monomial ideal I and any monomial prime ideal P, \({\text {HS}}_i(I(P))\subseteq {\text {HS}}_i(I)(P)\) for all i, where I(P) is the monomial localization

In this paper, by using variational approach, Mountain Pass Theorem and Krasnoselskii’s genus theory, we show the existence and multiplicity of solutions for a Schrödinger–Kirchhoff type equation involving the fractional \(p\left( .,.\right)\)-Laplacian in fractional Sobolev space with variable exponent. We also establish a Bartsch–Wang type compact embedding theorem for fractional Sobolev space with

We find a semi-algebraic description of the Minkowski sum \(\mathcal {A}_{3,n}\) of n copies of the twisted cubic segment \(\{(t,t^2,t^3)\mid -1\le t\le 1\}\) for each integer \(n\ge 3\). These descriptions provide efficient membership tests for the sets \(\mathcal {A}_{3,n}\). These membership tests in turn can be used to resolve some instances of the underdetermined matrix moment problem, which was

We make a first geometric study of three varieties in \(\mathbb {C}^m\otimes \mathbb {C}^m\otimes \mathbb {C}^m\) (for each m), including the Zariski closure of the set of tight tensors, the tensors with continuous regular symmetry. Our motivation is to develop a geometric framework for Strassen’s asymptotic rank conjecture that the asymptotic rank of any tight tensor is minimal. In particular, we

In a certain Lipschitz domain \(\Omega \subset {\mathbb {R}}^n\), we establish the boundary Harnack principle for positive superharmonic functions satisfying the nonlinear differential inequality \(-\Delta u\le cu^p\), where \(c>0\) and \(1

We give lower bounds for Hodge numbers of smooth well formed Fano weighted complete intersections. In particular, we compute their Hodge level, that is, the maximal distance between non-trivial Hodge numbers in the same row of the Hodge diamond. This allows us to classify varieties whose Hodge numbers are like that of a projective space, of a curve, or of a Calabi–Yau variety of low dimension.

The Malliavin integration-by-parts formula is a key ingredient to develop stochastic analysis on the Wiener space. In this article we show that a suitable integration-by-parts formula also characterizes a wide class of Gaussian processes, the so-called Gaussian Fredholm processes.

Any \({\mathcal {S}} \in \mathfrak {sp}(1,{\mathbb {R}})\) induces canonically a derivation S of the Heisenberg Lie algebra \({\mathfrak {h}}\) and so, a semi-direct extension \(G_{{\mathcal {S}}}=H \rtimes \exp ({\mathbb {R}}S)\) of the Heisenberg Lie group H (Müller and Ricci in Invent Math 101: 545–582, 1990). We shall explicitly describe the connected, simply connected Lie group \(G_{{\mathcal

The proof of Theorem 3.2 in the paper contains an error

We provide a Hochster type formula for the local cohomology modules of binomial edge ideals. As a consequence we obtain a simple criterion for the Cohen–Macaulayness and Buchsbaumness of these ideals and we describe their Castelnuovo–Mumford regularity and their Hilbert series. Conca and Varbaro (Square-free Groebner degenerations, 2018) have recently proved a conjecture of Conca, De Negri and Gorla

Many years ago, Griego, Heath and Ruiz-Moncayo proved that it is possible to define realizations of a sequence of uniform transport processes that converges almost surely to the standard Brownian motion, uniformly on the unit time interval. In this paper we extend their results to the multi parameter case. We begin constructing a family of processes, starting from a set of independent standard Poisson

Grothendieck’s theorem asserts that every continuous linear operator from \(\ell _1\) to \(\ell _2\) is absolutely (1, 1)-summing. This kind of result is commonly called coincidence result. In this paper we investigate coincidence results in the multilinear setting, showing how the cotype of the spaces involved affect such results. The special role played by \(\ell _1\) spaces is also investigated

It is proved that a module M over a Noetherian local ring R of prime characteristic and positive dimension has finite flat dimension if \({\text {Tor}}_i^R({}^{e}\!R, M)=0\) for \({\text {dim}}\,R\) consecutive positive values of i and infinitely many e. Here \({}^{e}\!R\) denotes the ring R viewed as an R-module via the eth iteration of the Frobenius endomorphism. In the case R is Cohen–Macualay,

We characterize the (regular) holonomicity of Horn systems of differential equations under a hypothesis that captures the most widely studied classical hypergeometric systems.

The aim of this paper is to prove the duality of Triebel–Lizorkin spaces \( F_{1,q\left( \cdot \right) }^{\alpha \left( \cdot \right) }\). First, we prove the duality of associated sequence spaces. The result follows from the so-called \(\varphi \)-transform characterization in the sense of Frazier and Jawerth.

The aim of this paper is to study some properties of left translates of a square integrable function on the Heisenberg group. First, a necessary and sufficient condition for the existence of the canonical dual to a function \(\varphi \in L^{2}(\mathbb {R}^{2n})\) is obtained in the case of twisted shift-invariant spaces. Further, characterizations of \(\ell ^{2}\)-linear independence and the Hilbertian

Let \(\Omega \subset {\mathbb {R}}^N, N\ge 2\), be a bounded smooth domain. In this paper, we consider a class of fractional Laplacian problems of the form $$\begin{aligned} \left\{ \begin{array}{ll} (\Delta )^s_{p_1(x,.)}u(x)+(\Delta )^s_{p_2(x,.)}u(x) + |u|^{q(x)-2}u = \lambda V_1(x)|u(x)|^{r_1(x)-2}u(x) \\ \qquad - \mu V_2(x)|u(x)|^{r_2(x)-2}u(x) \hbox { in }\Omega , \\ u(x) = 0 \; \hbox { in }

In this paper, we characterize power bounded weighted composition operators on weighted Bergman spaces of strongly pseudoconvex bounded domains in \({\mathbb {C}}^n\). Also, we introduce the notion of power bounded below operators, then, for \(\alpha >0\), we characterize power bounded below composition operators on \({\mathcal {D}}_\alpha \), the weighted Dirichlet space on the unit disk of the complex