Almost everywhere convergence of the inverse Jacobi transform and endpoint results for a disc multiplier

Troels Roussau Johansen

Studia Mathematica, Tome 204 (2011), p. 101-137 / Harvested from The Polish Digital Mathematics Library

Résumé

The maximal operator S⁎ for the spherical summation operator (or disc multiplier) $S_{R}$ associated with the Jacobi transform through the defining relation $\hat{S_{R} f} (λ) = 1_{| λ | \leq R} f ̂ (t)$ for a function f on ℝ is shown to be bounded from $L^{p} (ℝ ₊, d μ)$ into $L^{p} (ℝ, d μ) + L ² (ℝ, d μ)$ for (4α + 4)/(2α + 3) < p ≤ 2. Moreover S⁎ is bounded from $L^{p ₀, 1} (ℝ ₊, d μ)$ into $L^{p ₀, \infty} (ℝ, d μ) + L ² (ℝ, d μ)$ . In particular ${S_{R} f (t)}_{R > 0}$ converges almost everywhere towards f, for $f \in L^{p} (ℝ ₊, d μ)$ , whenever (4α + 4)/(2α + 3) < p ≤ 2.

Publié le : 2011-01-01

Zbl 1236.43004

EUDML-ID : urn:eudml:doc:285477

@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm205-2-1,
     author = {Troels Roussau Johansen},
     title = {Almost everywhere convergence of the inverse Jacobi transform and endpoint results for a disc multiplier},
     journal = {Studia Mathematica},
     volume = {204},
     year = {2011},
     pages = {101-137},
     zbl = {1236.43004},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm205-2-1}
}

Troels Roussau Johansen. Almost everywhere convergence of the inverse Jacobi transform and endpoint results for a disc multiplier. Studia Mathematica, Tome 204 (2011) pp. 101-137. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm205-2-1/