# Sharp Lower bound estimates for vector-valued and matrix-valued multipliers in $L^p$

@article{Boros2011SharpLB, title={Sharp Lower bound estimates for vector-valued and matrix-valued multipliers in \$L^p\$}, author={Nicholas Boros and Alexander Volberg}, journal={arXiv: Classical Analysis and ODEs}, year={2011} }

We generalize the idea of a multiplier in two different ways and generalize a recent result of Geiss, Montomery-Smith and Saksman. First of all, we consider multipliers in the form of a vector acting on a scalar function. Using this technique we compute the sharp lower bound estimate for $L^p$ operator norm of a quadratic perturbation of the real part of the Ahlfors-Beurling operator. Secondly, we consider matrix-valued multipliers to obtain a new proof showing that the $L^p$ operator norm of…

## 2 Citations

### The $L^p-$Operator Norm of a Quadratic Perturbation of the Real Part of the Ahlfors--Beurling Operator

- Mathematics
- 2011

We will explain how to compute the exact $L^p$ operator norm of a "quadratic perturbation" of the real part of the Ahlfors--Beurling operator. For the lower bound estimate we use a new approach of…

## References

SHOWING 1-10 OF 14 REFERENCES

### On singular integral and martingale transforms

- Mathematics
- 2007

Linear equivalences of norms of vector-valued singular integral operators and vector-valued martingale transforms are studied. In particular, it is shown that the UMD-constant of a Banach space X…

### Bellman function for the estimates of Littlewood-Paley type and asymptotic estimates in the p - 1 problem

- Mathematics
- 2005

### Classical and modern Fourier analysis

- Mathematics
- 2003

Prolegomena. 1. L p Spaces and Interpolation. 2. Maximal Functions, Fourier Transform, and Distributions. 3. Fourier Analysis on the Torus. 4. Singular Integrals of Convolution Type. 5.…

### Some remarks on Banach spaces in which martingale difference sequences are unconditional

- Mathematics
- 1983

### Heat extension of the Beurling operator and estimates for its norm , ( Russian . Russian summary ) , Algebra i Analiz 15 ( 2003 ) , 142 - 158 ; translation in St

- Ann . Acad . Sci . Fenn . Series AI Math .
- 1965

### Lp−bounds for a perturbation of Burkhold ers Martingale Transform

- 2011

### Heat extension of the Beurling operator and estimates for its norm, (Russian. Russian summary)

- Algebra i Analiz
- 2003

### Sharp L p −bounds for a small perturbation of Burkholders martingale transform

- 2011