log-Holder Continuous Condition Research Articles

Nonlinear unilateral problems without sign condition in Musielak spaces

We study the existence of solutions for some strongly nonlinear elliptic unilateral problems with $$L^1$$ data and without sign condition on the non-linearity, in the context of Musielak spaces, assuming essentially the log-Holder continuity condition on the Musielak function.

Isomorphisms of variable Hardy spaces associated with Schrödinger operators

Let L:= −Δ + V be the Schrodinger operator on ℝn with n ≥ 3, where V is a non-negative potential satisfying Δ−1 (V) ∈ L∞(ℝn). Let w be an L-harmonic function, determined by V, satisfying that there exists a positive constant o such that, for any x ∈ ℝn, 0 < δ ≤ w(x) ≤ 1. Assume that p(·): ℝn → (0, 1] is a variable exponent satisfying the globally log-Holder continuous condition. In this article, the authors show that the mappings $$H_L^{p\left( \cdot \right)}\left( {{\mathbb{R}^n}} \right)f \mapsto wf \in {H^{p\left( \cdot \right)}}\left( {{\mathbb{R}^n}} \right)$$ and $$H_L^{p\left( \cdot \right)}\left( {{\mathbb{R}^n}} \right)f \mapsto {\left( { - {\rm{\Delta }}} \right)^{1/2}}{L^{ - 1/2}}\left( f \right) \in {H^{p\left( \cdot \right)}}\left( {{\mathbb{R}^n}} \right)$$ are isomorphisms between the variable Hardy spaces (ℝn), associated with L, and the variable Hardy spaces Hp(·)(ℝn).

Molecular Decomposition of Anisotropic Hardy Spaces With Variable Exponents

Let A be an expansive dilation on ℝn, and p(·): ℝn → (0, ∞) be a variable exponent function satisfying the globally log-Holder continuous condition. Let \(H_A^{p\left(\cdot \right)}\left({{\mathbb{R}^n}} \right)\) be the variable anisotropic Hardy space defined via the non-tangential grand maximal function. In this paper, the authors establish its molecular decomposition, which is still new even in the classical isotropic setting (in the case A:= 2In×n). As applications, the authors obtain the boundedness of anisotropic Calderon-Zygmund operators from \(H_A^{p\left(\cdot \right)}\left({{\mathbb{R}^n}} \right)\) to Lp(·)(ℝn) or from \(H_A^{p\left(\cdot \right)}\left({{\mathbb{R}^n}} \right)\) to itself.

Molecular characterizations of variable anisotropic Hardy spaces with applications to boundedness of Calderón–Zygmund operators

Let $$p(\cdot ):\ \mathbb {R}^n\rightarrow (0,\infty ]$$ be a variable exponent function satisfying the globally log-Holder continuous condition and A a general expansive matrix on $$\mathbb {R}^n$$ . Let $$H_A^{p(\cdot )}(\mathbb {R}^n)$$ be the variable anisotropic Hardy space associated with A defined via the non-tangential grand maximal function. In this article, via the known atomic characterization of $$H_A^{p(\cdot )}(\mathbb {R}^n)$$ , the author establishes its molecular characterization with the known best possible decay of molecules. As an application, the author obtains a criterion on the boundedness of linear operators on $$H_A^{p(\cdot )}(\mathbb {R}^n)$$ , which is used to prove the boundedness of anisotropic Calderon–Zygmund operators on $$H_A^{p(\cdot )}(\mathbb {R}^n)$$ . In addition, the boundedness of anisotropic Calderon–Zygmund operators from $$H_A^{p(\cdot )}(\mathbb {R}^n)$$ to the variable Lebesgue space $$L^{p(\cdot )}(\mathbb {R}^n)$$ is also presented. All these results are new even in the classical isotropic setting.

Variable Besov Spaces Associated with Heat Kernels

Let $$({\mathcal {X}},\rho , \mu )$$ be a space of homogeneous type. Suppose that $$p(\cdot ),\ q(\cdot ):\ {\mathcal {X}}\rightarrow (0,\infty ]$$ are such that both $$1/p(\cdot )$$ and $$1/q(\cdot )$$ satisfy the globally log-Holder continuous condition, and $$s(\cdot ):\ {\mathcal {X}}\rightarrow \mathbb R$$ is a bounded function satisfying the locally log-Holder continuous condition. In this article, the authors introduce the variable Besov space $$B_{p(\cdot ),q(\cdot )}^{s(\cdot ),L}({\mathcal {X}})$$ , associated with a nonnegative self-adjoint operator L whose heat kernels satisfy small time Gaussian upper bound estimates, the Holder continuity, and the Markov property, which is new even on the sphere and the ball of $$\mathbb R^d$$ . Equivalent characterizations of this space, in terms of Peetre maximal functions and the heat semigroup, are established. Moreover, under the additional assumptions that $$\mu $$ satisfies the reverse doubling condition and the non-collapsing condition, its frame characterization is obtained. When L is the Laplacian operator on $$\mathbb R^d$$ , this space coincides with the existing variable Besov space.

Some properties of Musielak spaces with only the log-Hölder continuity condition and application

In this paper, we prove a density and a duality results in Musielak spaces, as well as an inequality of type Poincare, assuming only the log-Holder continuity condition. We will apply these results to give in non reflexive Musielak spaces the existence of solutions for some nonlinear parabolic problems with no continuous lower order terms.

Existence of Solutions for Some Nonlinear Elliptic Equations in Musielak Spaces with Only the Log-Hölder Continuity Condition

We study the existence of solutions for some nonlinear elliptic problems with $$L^1$$ data in Musielak spaces; we assume that the Musielak function satisfies only the log-Holder continuity condition.

N-Body Localization for the Anderson Model with Strongly Mixing Correlated Random Potentials

This work establishes the exponential spectral localization and the strong dynamical localization for the multi-particle Anderson tight-binding model with correlated but strongly mixing random external potential. The results are obtained in an energy interval near the lower edge of the spectrum of the multi-particle random Hamiltonian. In particular the exponential decay of the eigenfunctions is proved in the max-norm and the dynamical localization in the Hilbert-Schmidt norm. The proofs need the conditional probability distribution of the random external stochastic processes to obey the uniform log-Holder continuity condition.

Littlewood–Paley and Finite Atomic Characterizations of Anisotropic Variable Hardy–Lorentz Spaces and Their Applications

Let $$p(\cdot ):\ {{\mathbb {R}}}^n\rightarrow (0,\infty ]$$ be a variable exponent function satisfying the globally log-Holder continuous condition, $$q\in (0,\infty ]$$ and A be a general expansive matrix on $${\mathbb {R}}^n$$ . Let $$H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)$$ be the anisotropic variable Hardy–Lorentz space associated with A defined via the radial grand maximal function. In this article, the authors characterize $$H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)$$ by means of the Littlewood–Paley g-function or the Littlewood–Paley $$g_\lambda ^*$$ -function via first establishing an anisotropic Fefferman–Stein vector-valued inequality on the variable Lorentz space $$L^{p(\cdot ),q}({\mathbb {R}}^n)$$ . Moreover, the finite atomic characterization of $$H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)$$ is also obtained. As applications, the authors then establish a criterion on the boundedness of sublinear operators from $$H^{p(\cdot ),q}_A({\mathbb {R}}^n)$$ into a quasi-Banach space. Applying this criterion, the authors show that the maximal operators of the Bochner–Riesz and the Weierstrass means are bounded from $$H^{p(\cdot ),q}_A({\mathbb {R}}^n)$$ to $$L^{p(\cdot ),q}({\mathbb {R}}^n)$$ and, as consequences, some almost everywhere and norm convergences of these Bochner–Riesz and Weierstrass means are also obtained. These results on the Bochner–Riesz and the Weierstrass means are new even in the isotropic case.

On the Regularity of a Boundary Point for the p(x)-Laplacian

The Dirichlet problem for the p(x)-Laplacian with a continuous boundary function is considered, and a sufficient condition is found for the continuity of its solution at a boundary point, assuming that the exponent p(x) satisfies the log-Holder continuity condition at this point.

Interpolation Between $$H^{p(\cdot )}({\mathbb {R}}^n)$$ H p ( · ) ( R n ) and $$L^\infty ({\mathbb {R}}^n)$$ L ∞ ( R n ) : Real Method

Let $$p(\cdot ):\ {\mathbb {R}}^n\rightarrow (0,\infty )$$ be a variable exponent function satisfying the globally log-Holder continuous condition. In this article, the authors first obtain a decomposition for any distribution of the variable weak Hardy space into “good” and “bad” parts and then prove the following real interpolation theorem between the variable Hardy space $$H^{p(\cdot )}({\mathbb {R}}^n)$$ and the space $$L^{\infty }({\mathbb {R}}^n)$$ : $$(H^{p(\cdot )}(\mathbb R^n),L^{\infty }({\mathbb {R}}^n))_{\theta ,\infty }= WH^{p(\cdot )/(1-\theta )}({\mathbb {R}}^n),\quad \mathrm{where}~\theta \in (0,1), \mathrm{and}$$ $$WH^{p(\cdot )/(1-\theta )}({\mathbb {R}}^n)$$ denotes the variable weak Hardy space. As an application, the variable weak Hardy space $$WH^{p(\cdot )}({\mathbb {R}}^n)$$ with $$p_-:=\mathop {\text {ess inf}}\limits _{x\in {{{\mathbb {R}}}^n}}p(x)\in (1,\infty )$$ is proved to coincide with the variable Lebesgue space $$WL^{p(\cdot )}({\mathbb {R}}^n)$$ .

Molecular characterizations and dualities of variable exponent Hardy spaces associated with operators

Let L be a linear operator on L 2 (R n ) generating an analytic semigroup {e tL }t�0 with kernels having pointwise upper bounds and p(·): R n → (0,1) be a variable exponent function satisfying the globally log-Holder continuous condition. In this article, the authors introduce the variable exponent Hardy space associated with the operator L, denoted by H p(·) L (R n ), and the

Positive kernel operators in $$L^{p(x)}$$ spaces

A characterization of a weight $$v$$ governing the boundedness/compactness of the weighted kernel operator $$K_v$$ in variable exponent Lebesgue spaces $$L^{p(\cdot )}$$ is established under the log-Holder continuity condition on exponents of spaces. The kernel operator involves, for example, weighted variable parameter fractional integral operators. The distance between $$K_v$$ and the class of compact integral operators acting from $$L^{p(\cdot )}$$ to $$L^{q(\cdot )}$$ (measure of non-compactness) is also estimated from above and below.

Weight characterization of the trace inequality for the generalized Riemann-Liouville transform in L^p(x) spaces

Necessary and sufficient conditions on a weight governing the trace inequality for the Riemann-Liouville transform with variable parameter Rα(x) in L p(x) spaces are established provided that p and q satisfy the log-Holder continuity condition. Weighted criteria for the compactness of Rα(x) from Lp(x) to L q(x) v are also derived. Mathematics subject classification (2010): 46E30, 47B34.