Carleson measures and multipliers of Dirichlet-type spaces

Authors:
Ron Kerman and Eric Sawyer

Journal:
Trans. Amer. Math. Soc. **309** (1988), 87-98

MSC:
Primary 30D55; Secondary 46E99

MathSciNet review:
957062

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Abstract: A function from onto itself is a Dirichlet weight if it is increasing, and . The corresponding Dirichlet-type space, , consists of those bounded holomorphic functions on such that is integrable with respect to Lebesgue measure on . We characterize in terms of a Carleson-type maximal operator the functions in the set of pointwise multipliers of , .

**[1]**A. Bonami and R. Johnson,*Tent spaces based on the Lorentz spaces*, Math. Nachr.**132**(1987), 81–99. MR**910045**, 10.1002/mana.19871320107**[2]**Yitzhak Katznelson,*An introduction to harmonic analysis*, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0248482****[3]**R. Kerman and E. Sawyer,*Weighted norm inequalities for potentials with applications to Schrödinger operators, Fourier transforms, and Carleson measures*, Bull. Amer. Math. Soc. (N.S.)**12**(1985), no. 1, 112–116. MR**766965**, 10.1090/S0273-0979-1985-15306-6**[4]**R. Kerman and E. Sawyer,*The trace inequality and eigenvalue estimates for Schrödinger operators*, Ann. Inst. Fourier (Grenoble)**36**(1986), no. 4, 207–228 (English, with French summary). MR**867921****[5]**V. G. Maz'ya and T. O. Shaposhnikova,*Theory of multipliers in spaces of differentiable functions*, Pitman, 1985.**[6]**Alexander Nagel, Walter Rudin, and Joel H. Shapiro,*Tangential boundary behavior of functions in Dirichlet-type spaces*, Ann. of Math. (2)**116**(1982), no. 2, 331–360. MR**672838**, 10.2307/2007064**[7]**Eric T. Sawyer,*A characterization of a two-weight norm inequality for maximal operators*, Studia Math.**75**(1982), no. 1, 1–11. MR**676801****[8]**David A. Stegenga,*Multipliers of the Dirichlet space*, Illinois J. Math.**24**(1980), no. 1, 113–139. MR**550655****[9]**Elias M. Stein,*Singular integrals and differentiability properties of functions*, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR**0290095****[10]**Gerald D. Taylor,*Multipliers on 𝐷_{𝛼}*, Trans. Amer. Math. Soc.**123**(1966), 229–240. MR**0206696**, 10.1090/S0002-9947-1966-0206696-6

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1988-0957062-1

Article copyright:
© Copyright 1988
American Mathematical Society