Embeddings between grand, small, and variable Lebesgue spaces

The Sampling Theorem in Variable Lebesgue Spaces

hold. The facts above are well-known as the classical Shannon sampling theorem initially proved by Ogura [10]. Ashino and Mandai [1] generalized the sampling theorem in Lebesgue spaces L0(R) for 1 < p0 < ∞. Their generalized sampling theorem is the following. Theorem 1.1 ([1]). Let r > 0 and 1 < p0 < ∞. Then for all f ∈ L 0(R) with supp f̂ ⊂ [−rπ, rπ], we have the norm inequality C p r ‖f‖Lp0(Rn...

Continuous wavelet transform in variable Lebesgue spaces

In the present note we investigate norm and almost everywhere convergence of the inverse continuous wavelet transform in the variable Lebesgue space. Mathematics Subject Classification (2010): Primary 42C40, Secondary 42C15, 42B08, 42A38, 46B15.

Decomposition of Lebesgue Spaces

On Embeddings between Classical Lorentz Spaces

Let p 2 (0; 1), let v be a weight on (0; 1) and let p (v) be the classical Lorentz space, determined by the norm kfk p (v) := (R 1 0 (f (t)) p v(t) dt) 1=p. When p 2 (1; 1), this space is known to be a Banach space if and only if v is non-increasing, while it is only equivalent to a Banach space if and only if p (v) = ? p (v), where kfk ? p (v) := (R 1 0 (f (t)) p v(t) dt) 1=p. We may thus conc...

Riesz and Wolff potentials and elliptic equations in variable exponent weak Lebesgue spaces ∗

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