Optimal regularity for nonlinear elliptic equations with righthand side measure in variable exponent spaces

Relative weakly compact sets and weak convergence in variable exponent Lebesgue spaces {L^{p(cdot )}(Omega )} for infinite measure spaces (Omega ,mu ) are characterized. Criteria recently obtained in [14] for finite measures are here extended to the infinite measure case. In particular, it is showed that the inclusions between variable exponent Lebesgue spaces for infinite measures are never L-weakly compact. A lattice isometric representation of {L^{p(cdot )}(Omega )} as a variable exponent space L^{q(cdot )}(0,1) is given.

We prove optimal integrability results for solutions of the $${p(\cdot )}$$ -Laplace equation in the scale of (weak) Lebesgue spaces. To obtain this, we show that variable exponent Riesz and Wolff potentials map $$L^1$$ to variable exponent weak Lebesgue spaces.

In this paper we develop the theory of variable exponent Hardy spaces associated with discrete Laplacians on infinite graphs. Our Hardy spaces are defined by square integrals, atomic and molecular decompositions. Also we study boundedness properties of Littlewood-Paley functions, Riesz transforms, and spectral multipliers for discrete Laplacians on variable exponent Hardy spaces.

In this work, we present a Hodge-type decomposition for variable exponent spaces of Clifford-valued functions, where one of the components is the kernel of the parabolic-type Dirac operator.

On some p(x)-quasilinear problem with right-hand side measure

We give a survey of a selection of recent results on weighted and non-weighted estimations of classical operators of Harmonic Analysis in variable exponent Lebesgue, Morrey and Hölder spaces, based on the talk presented at International Conference Analysis, PDEs and Applications on the occasion of the 70th birthday of Vladimir Maz'ya, Rome, June 30–July 3, 2008. We touch both the Euclidean case and the general setting within the frameworks of quasimetric measure spaces. Some of the presented results are new.

Lp(0,l)’den Lq(0,l)’e agirlikli kesirli maksimal operatorunun kompaktligi icin gerekli ve yeterli sartlar uzerinde calisildi

We introduce function spaces for the treatment of parabolic equations with variable exponents by means of the theory of monotone operators. We generalize classical results such as density of smooth functions and a formula for integration by parts to prove existence, uniqueness and L 2-continuity of weak solutions to parabolic equations involving elliptic operators A with p(τ, x)-structure, where p is a globally log-Hölder continuous variable exponent satisfying 1 < p − ≤ p + < ∞.

In this paper we introduce the variable exponent Hormander spaces and we study some of their properties. In particular, it is shown that \({{(\mathcal{B}_{p_{(\cdot)}}^{c}(\Omega))^{\prime}}}\) is isomorphic to \({{\mathcal{B}^{loc}_{\widetilde{p^\prime(\cdot)}(\Omega)}}}\) (Ω open set in \({{\mathbb{R}^n, p? > 1}}\) and the Hardy–Littlewood maximal operator M is bounded in \({L_p(\cdot))}\) extending a Hormander’s result to our context. As a consequence, a number of results on sequence space representations of variable exponent Hormander spaces are given.

In this paper we characterize the dual $$\bigl (\mathcal {B}^c_{p(\cdot )} (\Omega ) \bigr )'$$ of the variable exponent Hormander space $$\mathcal {B}^c_{p(\cdot )} (\Omega )$$ when the exponent $$p(\cdot )$$ satisfies the conditions $$0 < p^- \le p^+ \le 1$$ , the Hardy-Littlewood maximal operator $$M$$ is bounded on $$L_{p(\cdot )/p_0}$$ for some $$0 < p_0 < p^-$$ and $$\Omega $$ is an open set in $$\mathbb {R}^n$$ . It is shown that the dual $$\bigl (\mathcal {B}^c_{p(\cdot )} (\Omega ) \bigr )'$$ is isomorphic to the Hormander space $$\mathcal {B}^{\mathrm {loc}}_\infty (\Omega )$$ (this is the $$p^+ \le 1$$ counterpart of the isomorphism $$\bigl (\mathcal {B}^c_{p(\cdot )} (\Omega ) \bigr )' \simeq \mathcal {B}^{\mathrm {loc}}_{\widetilde{p'(\cdot )}} (\Omega )$$ , $$1 < p^- \le p^+ < \infty $$ , recently proved by the authors) and hence the representation theorem $$\bigl ( \mathcal {B}^c_{p(\cdot )} (\Omega ) \bigr )' \simeq l^{\mathbb {N}}_\infty $$ is obtained. Our proof relies heavily on the properties of the Banach envelopes of the steps of $$\mathcal {B}^c_{p(\cdot )} (\Omega )$$ and on the extrapolation theorems in the variable Lebesgue spaces of entire analytic functions obtained in a precedent paper. Other results for $$p(\cdot ) \equiv p$$ , $$0 < p < 1$$ , are also given (e.g. $$\mathcal {B}^c_p (\Omega )$$ does not contain any infinite-dimensional $$q$$ -Banach subspace with $$p < q \le 1$$ or the quasi-Banach space $$\mathcal {B}_p \cap \mathcal {E}'(Q)$$ contains a copy of $$l_p$$ when $$Q$$ is a cube in $$\mathbb {R}^n$$ ). Finally, a question on complex interpolation (in the sense of Kalton) of variable exponent Hormander spaces is proposed.

In this paper, we study a class of Kirchhoff-type double phase equations with sublinear terms and two parameters in the framework of variable exponent function spaces. By employing a variant of the critical point theorem due to G. Bonanno, we establish the existence of at least three distinct solutions to the problem. Our result extends and complements recent contributions concerning double phase equations in the superlinear setting.

Weak solutions of fractional double-phase Laplacian-like problems in variable exponent Musielak-Orlicz Sobolev spaces

We investigate the direct and inverse theorems for trigonometric polynomials in the Morrey space $${{\mathcal {M}}}^{p(\cdot ),\lambda (\cdot )}$$ with variable exponents. For this space, we obtain estimates of the K-functional in terms of the modulus of smoothness and the Bernstein type inequality for trigonometric polynomials.

In this paper, using the atomic decomposition of the Herz-Morrey-Hardy spaces with variable exponent, the wavelet characterization by means of a local version of the discrete tent spaces with variable exponent is established. As an application, the boundedness of the fractional integral operators from variable exponent Herz-Morrey-Hardy spaces into variable exponent Herz-Morrey spaces is obtained.

Generalized fractional integral operators on variable exponent Morrey spaces of an integral form