Denseness of the spaces $Φ_V$ of Lizorkin type in the mixed $L^{p̅}(ℝ^n)$-spaces

Stefan Samko

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Displaying similar documents to “Denseness of the spaces $Φ_{V}$ of Lizorkin type in the mixed $L^{p ̅} (ℝ^{n})$ -spaces”

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Integrability theorems for trigonometric series

Bruce Aubertin, John Fournier (1993)

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We show that, if the coefficients (an) in a series $a_{0} / 2 + \sum_{n = 1}^{\infty} a_{n} c o s (n t)$ tend to 0 as n → ∞ and satisfy the regularity condition that $\sum_{m = 0}^{\infty} {\sum_{j = 1}^{\infty} [\sum_{n = j 2^{m}}^{(j + 1) 2^{m} - 1} | a_{n} - a_{n + 1} |] ²}^{1 / 2} < \infty$ , then the cosine series represents an integrable function on the interval [-π,π]. We also show that, if the coefficients (bn) in a series $\sum_{n = 1}^{\infty} b_{n} s i n (n t)$ tend to 0 and satisfy the corresponding regularity condition, then the sine series represents an integrable function on [-π,π] if and only if $\sum_{n = 1}^{\infty} | b_{n} | / n < \infty$ . These conclusions were previously known to hold under stronger restrictions on the sizes...

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In 1970 C. W. Onneweer formulated a sufficient condition for a periodic W-continuous function to have a Walsh-Fourier series which converges uniformly to the function. In this paper we extend his results from single to double Walsh-Fourier series in a more general setting. We study the convergence of rectangular partial sums in $L^{p}$ -norm for some 1 ≤ p ≤ ∞ over the unit square [0,1) × [0,1). In case p = ∞, by $L^{p}$ we mean $C_{W}$ , the collection of uniformly W-continuous functions f(x, y), endowed...

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An integral criterion for being an $H^{1} (ℝ^{2})$ Fourier multiplier is proved. It is applied in particular to suitable regular functions which depend on the product of variables.

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We develop a spectral-theoretic harmonic analysis for an arbitrary UMD space X. Our approach utilizes the spectral decomposability of X and the multiplier theory for $L_{X}^{p}$ to provide on the space X itself analogues of the classical themes embodied in the Littlewood-Paley Theorem, the Strong Marcinkiewicz Multiplier Theorem, and the M. Riesz Property. In particular, it is shown by spectral integration that classical Marcinkiewicz multipliers have associated transforms acting on X. ...

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Estimates of Fourier transforms in Sobolev spaces

V. Kolyada (1997)

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We investigate the Fourier transforms of functions in the Sobolev spaces $W_{1}^{r_{1}, . . ., r_{n}}$ . It is proved that for any function $f \in W_{1}^{r_{1}, . . ., r_{n}}$ the Fourier transform f̂ belongs to the Lorentz space $L^{n / r, 1}$ , where $r = n {(\sum_{j = 1}^{n} 1 / r_{j})}^{- 1} \leq n$ . Furthermore, we derive from this result that for any mixed derivative $D^{s} f (f \in C_{0}^{\infty}, s = (s_{1}, . . ., s_{n}))$ the weighted norm $∥ {(D^{s} f)}^{\land} ∥_{L^{1} (w)} (w (ξ) = {| ξ |}^{- n})$ can be estimated by the sum of $L^{1}$ -norms of all pure derivatives of the same order. This gives an answer to a question posed by A. Pełczyński and M. Wojciechowski.

$(H_{p}, L_{p})$ -type inequalities for the two-dimensional dyadic derivative

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It is shown that the restricted maximal operator of the two-dimensional dyadic derivative of the dyadic integral is bounded from the two-dimensional dyadic Hardy-Lorentz space $H_{p, q}$ to $L_{p, q}$ (2/3 < p < ∞, 0 < q ≤ ∞) and is of weak type $(L_{1}, L_{1})$ . As a consequence we show that the dyadic integral of a ∞ function $f \in L_{1}$ is dyadically differentiable and its derivative is f a.e.

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On (C,1) summability of integrable functions with respect to the Walsh-Kaczmarz system

G. Gát (1998)

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Let G be the Walsh group. For $f \in L^{1} (G)$ we prove the a. e. convergence σf → f(n → ∞), where $σ_{n}$ is the nth (C,1) mean of f with respect to the Walsh-Kaczmarz system. Define the maximal operator $σ * f ≔ s u p_{n} | σ_{n} f | .$ We prove that σ* is of type (p,p) for all 1 < p ≤ ∞ and of weak type (1,1). Moreover, $∥ σ * f ∥_{1} \leq c ∥ | f | ∥_{H}$ , where H is the Hardy space on the Walsh group.

Displaying similar documents to “Denseness of the spaces Φ V of Lizorkin type in the mixed L p ̅ ( ℝ n ) -spaces”

Displaying similar documents to “Denseness of the spaces $Φ_{V}$ of Lizorkin type in the mixed $L^{p ̅} (ℝ^{n})$ -spaces”