Embeddings of Weighted Generalized Morrey Spaces Into Lebesgue Spaces on Fractal Sets

Abstract

We study embeddings of weighted local and consequently global generalized Morrey spaces defined on a quasi-metric measure set (X, d, μ) of general nature which may be unbounded, into Lebesgue spaces Ls(X), 1 ≤ s ≤ p < ∞. The main motivation for obtaining such an embedding is to have an embedding of non-separable Morrey space into a separable space. In the general setting of quasi-metric measure spaces and arbitrary weights we give a sufficient condition for such an embedding. In the case of radial weights related to the center of local Morrey space, we obtain an effective sufficient condition in terms of (fractional in general) upper Ahlfors dimensions of the set X. In the case of radial weights we also obtain necessary conditions for such embeddings of local and global Morrey spaces, with the use of (fractional in general) lower and upper Ahlfors dimensions. In the case of power-logarithmic-type weights we obtain a criterion for such embeddings when these dimensions coincide.

A note on contributions concerning nonseparable spaces with respect to signal processing within Bayesian frameworks

“…The embedding $$$ X\hookrightarrow E $$$ in this case holds under the conditions $$$$ q<p\kern0.30em \mathrm{and}\kern0.30em \beta >\frac{\lambda }{p}+n\left(\frac{1}{q}-\frac{1}{p}\right), $$$$ as derived from Samko 24 . Corollary 3.14…”

“…Remark Theorem 3.3 may be extended to the so called generalized Morrey spaces (see their definition, e.g., in Rafeiro et al 29 ) via the use of embedding between generalized weighted Morrey and Lebesgue spaces obtained in Samko 24 . Since generalized Morrey spaces provide more flexible characterisation of how behave the averages $$$ \frac{1}{\mid B\left(x,r\right)\mid}\underset{B\left(x,r\right)}{\int }{\left|{u}_0(y)\right|}^p\kern0.1em dy $$$, this may give more possibilities for realization of the approach of Theorem 3.1, in particular, the “gap” between the convergences in $$$ X $$$ and $$$ E $$$ may be made more narrow in comparison with Theorem 3.3.…”

“…Theorem 3.3 may be extended to the so called generalized Morrey spaces (see their definition, e.g., in Rafeiro et al 29 ) via the use of embedding between generalized weighted Morrey and Lebesgue spaces obtained in Samko. 24 Since generalized Morrey spaces provide more flexible characterisation of how behave the averages |u 0 (𝑦)| p d𝑦, this may give more possibilities for realization of the approach of Theorem 3.1, in particular, the "gap" between the convergences in X and E may be made more narrow in comparison with Theorem 3.3. Various operators of harmonic analysis, important for applications in PDEs were studied in such spaces.…”

“…and arbitrarily close to 𝜆 p , since q in (3.2) may be chosen arbitrarily close to p. Convergence in L q 𝛽 (R n )-norm becomes "less weak" when 𝛽 → 𝜆 p . Theorem 3.3 may be extended to the so called generalized Morrey spaces (see their definition, e.g., in Rafeiro et al ) via the use of embedding between generalized weighted Morrey and Lebesgue spaces obtained in Samko 24. Since generalized Morrey spaces provide more flexible characterisation of how behave the averages1 |B(x,r)| ∫ B(x,r)…”

Weighted Fractional Hardy Operators and their Commutators on Generalized Morrey Spaces over Quasi-Metric Measure Spaces

“…We refer for Morrey spaces on quasi-metric measure spaces, for instance, to [21], [29], [33], [35] and also the survey [25]. More on Morrey spaces and their applications can be found in the two-volume book [32] of Y. Sawano et al…”