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On some subspaces of Morrey-Sobolev spaces and boundedness of Riesz integrals

100%

Dosso M., Fofana I., Sanogo M.

Annales Polonici Mathematici

2013

tom 108

nr 2

133-153

For 1 ≤ q ≤ α ≤ p ≤ ∞, $(L^q,l^p)^{α}$ is a complex Banach space which is continuously included in the Wiener amalgam space $(L^q,l^p)$ and contains the Lebesgue space $L^{α}$. We study the closure $(L^q,l^p)^{α}_{c,0}$ in $(L^q,l^p)^{α}$ of the space 𝓓 of test functions (infinitely differentiable and with compact support in $ℝ^d$) and obtain norm inequalities for Riesz potential operators and Riesz transforms in these spaces. We also introduce the Sobolev type space $W¹((L^q,l^p)^{α})$ (a subspace of a Morrey-Sobolev space, but a superspace of the classical Sobolev space $W^{1,α}$) and obtain in it Sobolev inequalities and a Kondrashov-Rellich compactness theorem.

On Fourier asymptotics of a generalized Cantor measure

100%

Kpata B., Fofana I., Koua K.

Colloquium Mathematicum

2010

tom 119

nr 1

109-122

Let d be a positive integer and μ a generalized Cantor measure satisfying $μ = ∑_{j = 1}^{m} a_{j}μ∘S_{j}^{-1}$, where $0 < a_{j} < 1$, $∑_{j = 1}^{m}a_{j} = 1$, $S_{j} = ρR + b_{j}$ with 0 < ρ < 1 and R an orthogonal transformation of $ℝ^{d}$. Then ⎧1 < p ≤ 2 ⇒ ⎨$sup_{r>0} r^{d(1/α'-1/p')} (∫_{J_{x}^{r}} |μ̂(y)|^{p'}dy)^{1/p'} ≤ D₁ρ^{-d/α'}$, $x ∈ ℝ^{d}$, ⎩ p = 2 ⇒ inf_{r≥1} r^{d(1/α'-1/2)} (∫_{J₀^{r}}|μ̂(y)|² dy)^{1/2} ≥ D₂ρ^{d/α'}$, where $J_{x}^{r} = ∏_{i=1}^{d} (x_{i} - r/2,x_{i} + r/2)$, α' is defined by $ρ^{d/α'} = (∑_{j=1}^{m} a_{j}^{p})^{1/p}$ and the constants D₁ and D₂ depend only on d and p.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Colloquium Mathematicum

2 Fofana I.

1 Dosso M.

1 Koua K.

1 Kpata B.

1 Sanogo M.

1 2013

1 2010

Wyniki wyszukiwania

On some subspaces of Morrey-Sobolev spaces and boundedness of Riesz integrals

On Fourier asymptotics of a generalized Cantor measure