The constant K in (3) is independent of f. Typically, we have m = 2, and the spaces involved are weighted Lebesgue spaces. Thus, for example, in (2),X = L1(R+, dx), A1 = L2(R+, dx) and A2 = L2(R+, x2 dx).It should be mentioned that both (1) and (2) are sharp, in the sense that theconstant π2can not be replaced by any smaller constant without violating the inequality.Carlson type inequalities have been studied by many authors, and applied toseveral branches of mathematics. We will only discuss a few of the generalizations and applications here.In what follows, we discuss some historically important generalizations of the Carlson inequalities, and illustrate how they can be applied. We then give a brief description of the papers I–V included in this thesis. At the end of this descriptive chapter, a short historical note about the mathematician Fritz Carlson is included.

*Contents*

1. Carlson Type Inequalities

2. The Development and Some Applications

2.1. Hardy

2.2. Beurling

2.3. Kjellberg

2.4. Levin

2.5. Barza, Burenkov, Peˇcari´c and Persson

2.6. Application to Harmonic Analysis

2.7. Application to Interpolation Theory

2.8. Application to Optimal Sampling

3. Paper I

4. Paper II

5. Paper III

6. Paper IV

7. Paper V

8. A Historical Note on Fritz Carlson

Acknowledgement

References

Author: Larsson, Leo

Source: Uppsala University Library

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