Sobolev embeddings for Riesz potentials of functions in Morrey spaces of variable exponent

On the role of Riesz potentials in Poisson's equation and Sobolev embeddings

“…Thus, one can apply the preceding theorems to the derivatives of the potential and obtain a family of results for p in this range (for a fixed α). There are also a considerable number of results for Riesz potentials beyond the basic framework of the L p -spaces, for example, the consideration of potentials with variable exponent [13], potentials mapping on L p -spaces with variable exponent [3], potentials acting on functions where the underlying space is assumed to be metric [4], potentials acting on Morrey spaces of variable exponent [14], potentials acting on functions in general Orlicz spaces [15] or Musielak-OrliczMorrey spaces [16]. This list is by no means exhaustive, though gives an idea of some of the possible variations one can consider and obtain results analogous to those we have recorded here.…”

Sobolev and Trudinger type inequalities on grand Musielak-Orlicz-Morrey spaces

“…[2, Theorem 3.1.4 (c)]) has been also extended to function spaces as above; see [22], [23] for Morrey spaces of variable exponent, [11] for grand Morrey spaces of variable exponent and [20] for Musielak-Orlicz-Morrey spaces.…”

“…[2, Theorem 3.1.4 (b)]) has been extended to various function spaces. For Morrey spaces, Sobolev's inequality was studied in [1], [27], [5], [25], etc., for Morrey spaces of variable exponent in [3], [13], [14], [22], [23], etc., for grand Morrey spaces in [21] and [17], and also for grand Morrey spaces of variable exponent in [11]. Recently, Sobolev's inequality has been extended by the authors [19] to an inequality for general potentials of functions in Musielak-Orlicz-Morrey spaces.…”

Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces

“…In the case of constant α, there was also proved a boundedness theorem in the limiting case p(x) = n−λ (x) α , when the potential operator I α acts from L p(·),λ(·) into BMO. In [29] the maximal operator and potential operators were considered in a somewhat more general space, but under more restrictive conditions on p(x). P. Hästö in [18] used his new "local-to-global" approach to extend the result of [2] on the maximal operator to the case of the whole space R n .…”

“…Variable exponent Morrey spaces L p(·),λ(·) ( ), were introduced and studied in [2] and [29] in the Euclidean setting and in [21] in the setting of metric measure spaces, in case of bounded sets. In [2] there was proved the boundedness of the maximal operator in variable exponent Morrey spaces L p(·),λ(·) ( ) under the log-condition on p(·) and λ(·) and for potential operators, under the same log-condition and the assumptions inf x∈ α(x) > 0, sup x∈ [λ(x) + α(x)p(x)] < n, there was proved a Sobolev type L p(·),λ(·) → L q(·),λ(·) -theorem.…”