A Note on Sharp Embedding Theorems for Holomorphic Classes Based on Lorentz Spaces on A Unit Circle

We provide new sharp embedding theorems for holomorphic classes in the unit disk based on Lorentz classes on the unit circle.


Introduction
Let µ be a positive Borel measure on a unit disk D = {z : |z| < 1}, T = {z : |z| = 1} be as usual unit circle.Let also L p,q (T ), 0 < p, q ≤ ∞ be the Lorentz space on T (see [3]), and dm(z) the normalized Lebesgue measure on T and d m 2 (z) be normalized Lebesgue measure on D. Let also ∆ jk be the dyadic decomposition of D, that is (see [1]) and Let further H(D) be the class of all holomorphic functions on D. For 0 Throughout the paper we write C (sometimes with indexes) to denote a positive constant which might be different at each occurrence (even in chain of inequalities) but is independent of the functions or variables being discussed.We say A is equivalent to B, denoted by A ≈ B, if there exist two constants C 1 and The goal of this note is to find some new sharp embedding theorems for H p,q α,s space for s = p, q = ∞ case in unit disk.Note such and other classes of holomorphic functions on the unit disk based on Lorentz spaces on the unit circle were studied in recent papers of Marc Lengfield (see [4,5,6]).Sharp embedding theorems for different holomorphic classes in D are well-known in literature see for example [1,7].

Main Results
Theorem 2.1.Let µ be a positive Borel measure on Then the following conditions are equivalent: (c) sup if and only if Proof of Theorem 2.1.Note that the following estimates are true By subharmonicity of | f (z)| p (see [1], Lemma 2.5), we used the estimates max , where I * jk is an enlarged arc (see [1], Lemma 2.5).
Using the fact that p > q and that V = τ − 2, (see [3]), we have finally from above 1 Let us prove the reverse.We will use the fact that for α > 0, if w is a center of ∆ jk and z ∈ ∆ J k , which can be checked by direct calculation.Let β be big enough positive number.
where w is a center of ∆ j 0 ,k 0 .Using the fact that (1 [4,5]), we get for fixed j 0 , k 0 sup j,k 1 and also So for any fixed j 0 , k 0 we will have if our embedding holds then So our theorem is proved.
The proof of Theorem 2.2 is very similar to the proof of Theorem 2.1 and we omit it.The third assertion can be obtained from the following chain of estimates 1 for α > 0, 0 < p < ∞ and τ > 0. This condition on measure is sufficient.The reverse implication can be obtained as above using the following standard test function (1 − ρξw) 1/p+α/p , where w is the center of ∆ jk .
Remark 2.4.We note that the following similar assertion was given in [7] for p < 2 and in [6] for all p < q.Let µ be a positive measure, where DI is well-known Carleson box and Note that the proof of mentioned assertions in [6,7] is based on completely different ideas.