Convolution algebras with weighted rearrangement-invariant norm

• Autorzy: R. Kerman , E. Sawyer
• Języki publikacji: EN

Abstrakty

EN

Let X be a rearrangement-invariant space of Lebesgue-measurable functions on $ℝ^n$, such as the classical Lebesgue, Lorentz or Orlicz spaces. Given a nonnegative, measurable (weight) function on $ℝ^n$, define $X(w) = {F: ℝ^n → ℂ: ∞ > ∥F∥_{X(w)} := ∥Fw∥_X}$. We investigate conditions on such a weight w that guarantee X(w) is an algebra under the convolution product F∗G defined at $x ∈ ℝ^n$ by $(F∗G)(x) = ʃ_{ℝ^n} F(x-y)G(y)dy$; more precisely, when $∥F∗G∥_{X(w)} ≤ ∥F∥_{X(w)} ∥G∥_{X(w)}$ for all F,G ∈ X(w).

Kategorie tematyczne

• 42A85: Convolution, factorization
• 46E30: Spaces of measurable functions ( L p -spaces, Orlicz spaces, K\"othe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1994
• Tom: 108
• Numer: 2
• Strony: 103-126

Daty

wydano

1994

otrzymano

1992-01-07

poprawiono

1993-05-24

Twórcy

autor

R. Kerman

• Department of Mathematics, Brock University, St. Catharines, Ontario, Canada L2S 3A1

autor

E. Sawyer

• Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1

Bibliografia

• [1] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, 1988.
• [2] A. Beurling, Sur les intégrales de Fourier absolument convergentes et leur application à une transformation fonctionnelle, Neuvième Congrès Math. Scand., Helsingfors, 1938, 345-366.
• [3] D. Boyd, The Hilbert transform on rearrangement-invariant spaces, Canad. J. Math. 19 (1967), 599-616.
• [4] A. P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113-190.
• [5] M. Cwikel and P. Nilsson, Interpolation of weighted Banach lattices, Mem. Amer. Math. Soc., to appear.
• [6] T. A. Gillespie, Factorization in Banach function spaces, Nederl. Akad. Wetensch. Proc. Ser. A 84 (1981), 287-300.
• [7] S. Grabiner, Weighted convolution algebras on the half line, J. Math. Anal. Appl. 83 (1981), 531-583.
• [8] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, II, Springer, Berlin, 1963.
• [9] R. Howard and A. Schep, Norms of positive operators on $L^p$-spaces, Proc. Amer. Math. Soc. 109 (1990), 135-146.
• [10] E. Kerlin and A. Lambert, Strictly cyclic shifts on $l_p$, Acta Sci. Math. (Szeged) 35 (1973), 87-94.
• [11] G. Ya. Lozanovskiĭ, On some Banach lattices, Sibirsk. Mat. Zh. 10 (1969), 419-431 (in Russian).
• [12] N. K. Nikol'skiĭ, Invariant subspaces of the shift operator in some sequence spaces, Candidate's Dissertation, Leningrad, 1966 (in Russian).
• [13] F. Riesz, Sur une inégalité intégrale, J. London Math. Soc. 5 (1930), 162-168.
• [14] S. L. Sobolev, On a theorem of functional analysis, Mat. Sb. 46 (1938), 471-497 (in Russian).