Growth Envelopes of Some Variable and Mixed Function Spaces

We study unboundedness properties of functions belonging to Lebesgue and Lorentz spaces with variable and mixed norms using growth envelopes. Our results extend the ones for the corresponding classical spaces in a natural way. In the case of spaces with mixed norms, it turns out that the unboundedness in the worst direction, i.e., in the direction where $$p_{i}$$ p i is the smallest, is crucial. More precisely, the growth envelope is given by $${\mathfrak {E}}_{{\mathsf {G}}}(L_{\overrightarrow{p}}(\varOmega )) = (t^{-1/\min \{p_{1}, \ldots , p_{d} \}},\min \{p_{1}, \ldots , p_{d} \})$$ E G ( L p → ( Ω ) ) = ( t - 1 / min { p 1 , … , p d } , min { p 1 , … , p d } ) for mixed Lebesgue and $${\mathfrak {E}}_{{\mathsf {G}}}(L_{\overrightarrow{p},q}(\varOmega )) = (t^{-1/\min \{p_{1}, \ldots , p_{d} \}},q)$$ E G ( L p → , q ( Ω ) ) = ( t - 1 / min { p 1 , … , p d } , q ) for mixed Lorentz spaces, respectively. For the variable Lebesgue spaces, we obtain $${\mathfrak {E}}_{{\mathsf {G}}}(L_{p(\cdot )}(\varOmega )) = (t^{-1/p_{-}},p_{-})$$ E G ( L p ( · ) ( Ω ) ) = ( t - 1 / p - , p - ) , where $$p_{-}$$ p - is the essential infimum of $$p(\cdot )$$ p ( · ) , subject to some further assumptions. Similarly, for the variable Lorentz space, it holds$${\mathfrak {E}}_{{\mathsf {G}}}(L_{p(\cdot ),q}(\varOmega )) = (t^{-1/p_{-}},q)$$ E G ( L p ( · ) , q ( Ω ) ) = ( t - 1 / p - , q ) . The growth envelope is used for Hardy-type inequalities and limiting embeddings. In particular, as a by-product, we determine the smallest classical Lebesgue (Lorentz) space which contains a fixed mixed or variable Lebesgue (Lorentz) space, respectively.

Regularity of Weak Solutions to Elliptic Problem with Irregular Data

This paper is concerned with the study of the nonlinear elliptic equations in a bounded subset Ω ⊂ RN Au = f, where A is an operator of Leray-Lions type acted from the space W1,p(·)0(Ω) into its dual. when the second term f belongs to Lm(·), with m(·) > 1 being small. we prove existence and regularity of weak solutions for this class of problems p(x)-growth conditions. The functional framework involves Sobolev spaces with variable exponents as well as Lebesgue spaces with variable exponents.

Weighted and endpoint estimates for commutators of bilinear pseudo-differential operators

<abstract><p>In this paper, by applying the accurate estimates of the Hörmander class, the authors consider the commutators of bilinear pseudo-differential operators and the operation of multiplication by a Lipschitz function. By establishing the pointwise estimates of the corresponding sharp maximal function, the boundedness of the commutators is obtained respectively on the products of weighted Lebesgue spaces and variable exponent Lebesgue spaces with $ \sigma \in\mathcal{B}BS_{1, 1}^{1} $. Moreover, the endpoint estimate of the commutators is also established on $ L^{\infty}\times L^{\infty} $.</p></abstract>

The embedding theorems for anisotropic Nikol’skii-Besov spaces with generalized mixed smoothness

The theory of embedding of spaces of differentiable functions studies the important relations of differential (smoothness) properties of functions in various metrics and has a wide application in the theory of boundary value problems of mathematical physics, approximation theory, and other fields of mathematics. In this article, we prove the embedding theorems for anisotropic spaces Nikol’skii-Besov with a generalized mixed smoothness and mixed metric, and anisotropic Lorentz spaces. The proofs of the obtained results are based on the inequality of different metrics for trigonometric polynomials in Lebesgue spaces with mixed metrics and interpolation properties of the corresponding spaces.

Локальные гранд пространства Лебега

We introduce ``local grand'' Lebesgue spaces $L^{p),\theta}_{x_0,a}(\Omega)$, $0<p<\infty,$ $\Omega \subseteq \mathbb{R}^n$, where the process of ``grandization'' relates to a single point $x_0\in \Omega$, contrast to the case of usual known grand spaces $L^{p),\theta}(\Omega)$, where ``grandization'' relates to all the points of $\Omega$. We define the space $L^{p),\theta}_{x_0,a}(\Omega)$ by means of the weight $a(|x-x_0|)^{\varepsilon p}$ with small exponent, $a(0)=0$. Under some rather wide assumptions on the choice of the local ``grandizer'' $a(t)$, we prove some properties of these spaces including their equivalence under different choices of the grandizers $a(t)$ and show that the maximal, singular and Hardy operators preserve such a ``single-point grandization'' of Lebesgue spaces $L^p(\Omega)$, $1<p<\infty$, provided that the lower Matuszewska--Orlicz index of the function $a$ is positive. A Sobolev-type theorem is also proved in local grand spaces under the same condition on the grandizer.