In this paper, we continue the study of Lebesgue-type inequalities for greedy algorithms. We introduce notion strong partially Markushevich bases and parameters associated with them. prove that property is equivalent to being conservative quasi-greedy, extending a similar result given in Dilworth et al. (Constr Approx 19:575–597, 2003) Schauder bases. also give characterization 1-strong partial...

We define a Muckenhoup-type condition on weighted Morrey spaces using the Köthe dual of space. show that is necessary and sufficient for boundedness maximal operator defined with balls centered at origin spaces. A modified characterizes inequalities Calderón operator. also local usual Hardy–Littlewood operator, simplifying previous characterization Nakamura–Sawano–Tanaka. For same in case globa...

It is well-known that a random variable, i.e. a function defined on a probability space, with values in a Borel space, can be represented on the special probability space consisting of the unit interval with Lebesgue measure. We show an extension of this to multivariate functions. This is motivated by some recent constructions of random

Probability enjoys a monadic structure (Lawvere 1962; Giry 1981; Ramsey and Pfeffer 2002). A monadic computation represents a probability distribution, and the unit operation return a creates the (Dirac) distribution “certainly a.” The bind operation combines a distribution of type M a and a function of type a -> M b; the function is a probability kernel (Pollard 2002), and it represents the co...

Profile decompositions for “critical” Sobolev-type embeddings are established, allowing one to regain some compactness despite the non-compact nature of the embeddings. Such decompositions have wide applications to the regularity theory of nonlinear partial differential equations, and have typically been established for spaces with Hilbert structure. Following the method of S. Jaffard, we treat...

In this paper we prove theorems on the existence of integrable and monotonic solutions of nonlinear integral equation in Lebesgue Space. The basic tool used in the proof is the fixed point theorem due to Darbo with respect to the so-called measure of weak noncompactness.

We introduce local grand Lebesgue spaces, over a quasi-metric measure space \( ( X,d, \mu ) \), where the is “aggrandized” not everywhere but only at given closed set F of zero. show that such spaces coincide for different choices aggrandizers if their Matuszewska–Orlicz indices are positive. Within framework we study maximal operator, singular operators with standard kernel, and potential type...

To motivate the elaborate machinery of measure theory, it is desirable to have an example of a set which is not measurable in some natural space. The usual example is the Vitali set, obtained by picking one representative from each equivalence class of R induced by the relation x ∼ y iff x − y ∈ Q. The translation-invariance of Lebesgue measure implies that the resulting set is not Lebesgue-mea...

For the 3D Navier–Stokes problem on the whole space, we study existence, regularity and stability of time-periodic solutions in Lebesgue, Lorentz or Sobolev spaces, when the periodic forcing belongs to critical classes of forces.