Washington University in St. Louis

PHD Student

I have been a teaching assistant and research assistant.

Georgia Institute of Technology

Teaching Assistant

I assist the lead instructor in the instruction of students. I teach two classes a week. I am responsible for making my lesson plans, grading exams and helping students in office hours.

Texas A&M University

Visiting Assistant Professor

Robert worked at Texas A&M University as a Visiting Assistant Professor

Durham Geo Enterprises

Assembly Technician

Assemble ground-water remediation pumps, material s testing equipment and geotechnical instrumentation. Perform analysis and repairs on customers' equipment. Work with management and co-workers to develop testing and verification procedures.

Weighted Estimates for the Berezin Transform and Bergman Projection on the Unit Ball in ℂn

Accepted to Math. Z.

Using modern techniques of dyadic harmonic analysis, we are able to prove sharp estimates for the Bergman projection and Berezin transform and more general operators in weighted Bergman spaces on the unit ball in ℂn. The estimates are in terms of the Bekolle-Bonami constant of the weight.

Weighted Estimates for the Berezin Transform and Bergman Projection on the Unit Ball in ℂn

Accepted to Math. Z.

Using modern techniques of dyadic harmonic analysis, we are able to prove sharp estimates for the Bergman projection and Berezin transform and more general operators in weighted Bergman spaces on the unit ball in ℂn. The estimates are in terms of the Bekolle-Bonami constant of the weight.

Entropy Bumps and Another Sufficient Condition for the Two--Weight Boundedness of Sparse Operators

Accepted to Israel Journal of Math

In this short note, we give a very efficient proof of a recent result of Treil-Volberg and Lacey--Spencer giving sufficient conditions for the two-weight boundedness of a sparse operator. We also give a new sufficient condition for the two-weight boundedness of a sparse operator. We make critical use of a formula of Hytonen

Weighted Estimates for the Berezin Transform and Bergman Projection on the Unit Ball in ℂn

Accepted to Math. Z.

Using modern techniques of dyadic harmonic analysis, we are able to prove sharp estimates for the Bergman projection and Berezin transform and more general operators in weighted Bergman spaces on the unit ball in ℂn. The estimates are in terms of the Bekolle-Bonami constant of the weight.

Entropy Bumps and Another Sufficient Condition for the Two--Weight Boundedness of Sparse Operators

Accepted to Israel Journal of Math

In this short note, we give a very efficient proof of a recent result of Treil-Volberg and Lacey--Spencer giving sufficient conditions for the two-weight boundedness of a sparse operator. We also give a new sufficient condition for the two-weight boundedness of a sparse operator. We make critical use of a formula of Hytonen

Ap weights and Quantitative Estimates in the Schroedinger Setting

Submitted to Transactions of the AMS

Suppose L=−Δ+V is a Schr\"odinger operator on ℝn with a potential V belonging to certain reverse H\"older class RHσ with σ≥n/2. The aim of this paper is to study the Ap weights associated to L, denoted by ALp, which is a larger class than the classical Muckenhoupt Ap weights. We first establish the "exp--log" link between ALp and BMOL (the BMO space associated with L), which is the first extension of the classical result to a setting beyond the Laplace operator. Second, we prove the quantitative ALp bound for the maximal function and the maximal heat semigroup associated to L. Then we further provide the quantitative ALp,q bound for the fractional integral operator associated to L. We point out that all these quantitative bounds are known before in terms of the classical Ap,q constant. However, since Ap,q⊂ALp,q, the ALp,q constants are smaller than Ap,q constant. Hence, our results here provide a better quantitative constant for maximal functions and fractional integral operators associated to L.

Weighted Estimates for the Berezin Transform and Bergman Projection on the Unit Ball in ℂn

Accepted to Math. Z.

Using modern techniques of dyadic harmonic analysis, we are able to prove sharp estimates for the Bergman projection and Berezin transform and more general operators in weighted Bergman spaces on the unit ball in ℂn. The estimates are in terms of the Bekolle-Bonami constant of the weight.

Entropy Bumps and Another Sufficient Condition for the Two--Weight Boundedness of Sparse Operators

Accepted to Israel Journal of Math

In this short note, we give a very efficient proof of a recent result of Treil-Volberg and Lacey--Spencer giving sufficient conditions for the two-weight boundedness of a sparse operator. We also give a new sufficient condition for the two-weight boundedness of a sparse operator. We make critical use of a formula of Hytonen

Ap weights and Quantitative Estimates in the Schroedinger Setting

Submitted to Transactions of the AMS

Suppose L=−Δ+V is a Schr\"odinger operator on ℝn with a potential V belonging to certain reverse H\"older class RHσ with σ≥n/2. The aim of this paper is to study the Ap weights associated to L, denoted by ALp, which is a larger class than the classical Muckenhoupt Ap weights. We first establish the "exp--log" link between ALp and BMOL (the BMO space associated with L), which is the first extension of the classical result to a setting beyond the Laplace operator. Second, we prove the quantitative ALp bound for the maximal function and the maximal heat semigroup associated to L. Then we further provide the quantitative ALp,q bound for the fractional integral operator associated to L. We point out that all these quantitative bounds are known before in terms of the classical Ap,q constant. However, since Ap,q⊂ALp,q, the ALp,q constants are smaller than Ap,q constant. Hence, our results here provide a better quantitative constant for maximal functions and fractional integral operators associated to L.

Essential Norm of Operators on Vector Valued Bergman Space

Contemporary Mathematics: Function Spaces in Analysis, vol. 645.

Weighted Estimates for the Berezin Transform and Bergman Projection on the Unit Ball in ℂn

Accepted to Math. Z.

Using modern techniques of dyadic harmonic analysis, we are able to prove sharp estimates for the Bergman projection and Berezin transform and more general operators in weighted Bergman spaces on the unit ball in ℂn. The estimates are in terms of the Bekolle-Bonami constant of the weight.

Entropy Bumps and Another Sufficient Condition for the Two--Weight Boundedness of Sparse Operators

Accepted to Israel Journal of Math

In this short note, we give a very efficient proof of a recent result of Treil-Volberg and Lacey--Spencer giving sufficient conditions for the two-weight boundedness of a sparse operator. We also give a new sufficient condition for the two-weight boundedness of a sparse operator. We make critical use of a formula of Hytonen

Ap weights and Quantitative Estimates in the Schroedinger Setting

Submitted to Transactions of the AMS

Suppose L=−Δ+V is a Schr\"odinger operator on ℝn with a potential V belonging to certain reverse H\"older class RHσ with σ≥n/2. The aim of this paper is to study the Ap weights associated to L, denoted by ALp, which is a larger class than the classical Muckenhoupt Ap weights. We first establish the "exp--log" link between ALp and BMOL (the BMO space associated with L), which is the first extension of the classical result to a setting beyond the Laplace operator. Second, we prove the quantitative ALp bound for the maximal function and the maximal heat semigroup associated to L. Then we further provide the quantitative ALp,q bound for the fractional integral operator associated to L. We point out that all these quantitative bounds are known before in terms of the classical Ap,q constant. However, since Ap,q⊂ALp,q, the ALp,q constants are smaller than Ap,q constant. Hence, our results here provide a better quantitative constant for maximal functions and fractional integral operators associated to L.

Essential Norm of Operators on Vector Valued Bergman Space

Contemporary Mathematics: Function Spaces in Analysis, vol. 645.

The Essential Norm of Operators on $\ell^2$--Valued Bergman--Type Function Spaces

Complex Analysis and Operator Theory

In this paper we consider the reproducing kernel thesis for boundedness and compactness for operators on ℓ2--valued Bergman-type spaces. This paper generalizes many well--known results about classical function spaces to their ℓ2--valued versions. In particular, the results in this paper apply to the weighted ℓ2--valued Bergman space on the unit ball, the unit polydisc and, more generally to weighted Fock spaces.

Weighted Estimates for the Berezin Transform and Bergman Projection on the Unit Ball in ℂn

Accepted to Math. Z.

Using modern techniques of dyadic harmonic analysis, we are able to prove sharp estimates for the Bergman projection and Berezin transform and more general operators in weighted Bergman spaces on the unit ball in ℂn. The estimates are in terms of the Bekolle-Bonami constant of the weight.

Entropy Bumps and Another Sufficient Condition for the Two--Weight Boundedness of Sparse Operators

Accepted to Israel Journal of Math

In this short note, we give a very efficient proof of a recent result of Treil-Volberg and Lacey--Spencer giving sufficient conditions for the two-weight boundedness of a sparse operator. We also give a new sufficient condition for the two-weight boundedness of a sparse operator. We make critical use of a formula of Hytonen

Ap weights and Quantitative Estimates in the Schroedinger Setting

Submitted to Transactions of the AMS

Suppose L=−Δ+V is a Schr\"odinger operator on ℝn with a potential V belonging to certain reverse H\"older class RHσ with σ≥n/2. The aim of this paper is to study the Ap weights associated to L, denoted by ALp, which is a larger class than the classical Muckenhoupt Ap weights. We first establish the "exp--log" link between ALp and BMOL (the BMO space associated with L), which is the first extension of the classical result to a setting beyond the Laplace operator. Second, we prove the quantitative ALp bound for the maximal function and the maximal heat semigroup associated to L. Then we further provide the quantitative ALp,q bound for the fractional integral operator associated to L. We point out that all these quantitative bounds are known before in terms of the classical Ap,q constant. However, since Ap,q⊂ALp,q, the ALp,q constants are smaller than Ap,q constant. Hence, our results here provide a better quantitative constant for maximal functions and fractional integral operators associated to L.

Essential Norm of Operators on Vector Valued Bergman Space

Contemporary Mathematics: Function Spaces in Analysis, vol. 645.

The Essential Norm of Operators on $\ell^2$--Valued Bergman--Type Function Spaces

Complex Analysis and Operator Theory

In this paper we consider the reproducing kernel thesis for boundedness and compactness for operators on ℓ2--valued Bergman-type spaces. This paper generalizes many well--known results about classical function spaces to their ℓ2--valued versions. In particular, the results in this paper apply to the weighted ℓ2--valued Bergman space on the unit ball, the unit polydisc and, more generally to weighted Fock spaces.

Some Entropy Bump Conditions for Fractional Maximal and Integral Operators

Concrete Operators

We investigate weighted inequalities for fractional maximal operators and fractional integral operators. We work within the innovative framework of "entropy bounds" introduced by Treil--Volberg. Using techniques developed by Lacey and the second author, we are able to efficiently prove the weighted inequalities.

Weighted Estimates for the Berezin Transform and Bergman Projection on the Unit Ball in ℂn

Accepted to Math. Z.

Using modern techniques of dyadic harmonic analysis, we are able to prove sharp estimates for the Bergman projection and Berezin transform and more general operators in weighted Bergman spaces on the unit ball in ℂn. The estimates are in terms of the Bekolle-Bonami constant of the weight.

Entropy Bumps and Another Sufficient Condition for the Two--Weight Boundedness of Sparse Operators

Accepted to Israel Journal of Math

In this short note, we give a very efficient proof of a recent result of Treil-Volberg and Lacey--Spencer giving sufficient conditions for the two-weight boundedness of a sparse operator. We also give a new sufficient condition for the two-weight boundedness of a sparse operator. We make critical use of a formula of Hytonen

Ap weights and Quantitative Estimates in the Schroedinger Setting

Submitted to Transactions of the AMS

Suppose L=−Δ+V is a Schr\"odinger operator on ℝn with a potential V belonging to certain reverse H\"older class RHσ with σ≥n/2. The aim of this paper is to study the Ap weights associated to L, denoted by ALp, which is a larger class than the classical Muckenhoupt Ap weights. We first establish the "exp--log" link between ALp and BMOL (the BMO space associated with L), which is the first extension of the classical result to a setting beyond the Laplace operator. Second, we prove the quantitative ALp bound for the maximal function and the maximal heat semigroup associated to L. Then we further provide the quantitative ALp,q bound for the fractional integral operator associated to L. We point out that all these quantitative bounds are known before in terms of the classical Ap,q constant. However, since Ap,q⊂ALp,q, the ALp,q constants are smaller than Ap,q constant. Hence, our results here provide a better quantitative constant for maximal functions and fractional integral operators associated to L.

Essential Norm of Operators on Vector Valued Bergman Space

Contemporary Mathematics: Function Spaces in Analysis, vol. 645.

The Essential Norm of Operators on $\ell^2$--Valued Bergman--Type Function Spaces

Complex Analysis and Operator Theory

In this paper we consider the reproducing kernel thesis for boundedness and compactness for operators on ℓ2--valued Bergman-type spaces. This paper generalizes many well--known results about classical function spaces to their ℓ2--valued versions. In particular, the results in this paper apply to the weighted ℓ2--valued Bergman space on the unit ball, the unit polydisc and, more generally to weighted Fock spaces.

Some Entropy Bump Conditions for Fractional Maximal and Integral Operators

Concrete Operators

We investigate weighted inequalities for fractional maximal operators and fractional integral operators. We work within the innovative framework of "entropy bounds" introduced by Treil--Volberg. Using techniques developed by Lacey and the second author, we are able to efficiently prove the weighted inequalities.

Two-Weight Inequalities for Commutators with Fractional Integral Operators

Studia Math.

In this paper we investigate weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a function. In particular, we show that, for μ,λ∈Ap,q and α/n+1/q=1/p, the norm ∥[b,Iα]:Lp(μp)→Lq(λq)∥ is equivalent to the norm of b in the weighted BMO space BMO(ν), where ν=μλ−1. This work extends some of the results on this topic existing in the literature, and continues a line of investigation which was initiated by Bloom in 1985 and was recently developed further by the first author, Lacey, and Wick.