Fejér-Riesz Inequality For H P H^p H P On The Unit Disk For More General Curves

Introduction

The Fejér-Riesz inequality is a fundamental result in the theory of Hardy spaces, which provides an upper bound for the $L^p$ norm of a function on the unit disk. This inequality has far-reaching implications in various areas of mathematics, including complex analysis, harmonic analysis, and partial differential equations. In this article, we will discuss the Fejér-Riesz inequality for $H^p$ on the unit disk and its generalization to more general curves.

Hardy Spaces and the Fejér-Riesz Inequality

The Hardy space $H^p$ on the unit disk is a Banach space of holomorphic functions on the unit disk $\mathbb{D}$, equipped with the norm ${ \|f\|_{H^p} = \sup_{0 < r < 1} \left( \int_{\partial \mathbb{D}} |f(re^{i\theta})|^p \, d\sigma(\theta) \right)^{1/p}, }$ where $d\sigma$ is the normalized Lebesgue measure on the unit circle $\partial \mathbb{D}$.

The Fejér-Riesz inequality states that for any function $f \in H^p$, we have ${ \int_{-1}^{1} |f(x)|^p \, dx \leq \frac{1}{2} \int_{-1}^{1} |f'(x)|^p \, dx. }$ This inequality provides a useful estimate for the $L^p$ norm of a function on the unit disk, which is essential in various applications.

Proof of the Fejér-Riesz Inequality

The proof of the Fejér-Riesz inequality is based on the following key idea: we can represent the function $f$ as a Poisson integral of its boundary values. Specifically, we can write ${ f(z) = \frac{1}{2\pi} \int_{-1}^{1} \frac{1 - |z|^2}{|1 - z\overline{w}|^2} f(w) \, dw, }$ where $z \in \mathbb{D}$ and $w \in [-1, 1]$.

Using this representation, we can estimate the $L^p$ norm of $f$ as follows: ${ \int_{-1}^{1} |f(x)|^p \, dx \leq \frac{1}{2\pi} \int_{-1}^{1} \int_{-1}^{1} \frac{1 - |x|^2}{|1 - x\overline{w}|^2} |f(w)|^p \, dw \, dx. }$ By applying the Hölder inequality, we can bound the above expression by ${ \frac{1}{2\pi} \int_{-1}^{1} \left( \int_{-1}^{1} \frac{1 - |x|^2}{|1 - x\overline{w}|^2} \, dx \right)^{1/p'} |f(w)|^p \, dw, }$ where $p'$ is the conjugate exponent of $p$.

Using the fact that the Poisson kernel is subharmonic, we can show that the integral ${ \int_{-1}^{1} \frac{1 - |x|^2}{|1 - x\overline{w}|^2} \, dx }$ is bounded by a constant times $|f'(w)|^p$. This allows us to conclude that ${ \int_{-1}^{1} |f(x)|^p \, dx \leq \frac{1}{2} \int_{-1}^{1} |f'(x)|^p \, dx, }$ which is the desired Fejér-Riesz inequality.

Generalization to More General Curves

The Fejér-Riesz inequality can be generalized to more general curves, such as the unit circle in $\mathbb{C}^n$ or the boundary of a bounded domain in $\mathbb{C}^n$. In these cases, we need to modify the definition of the Hardy space $H^p$ and the Poisson kernel accordingly.

For example, if we consider the unit circle in $\mathbb{C}^n$, we can define the Hardy space $H^p$ as the space of holomorphic functions on the unit ball $\mathbb{B}^n$ that satisfy the growth condition ${ \|f\|_{H^p} = \sup_{0 < r < 1} \left( \int_{\partial \mathbb{B}^n} |f(re^{i\theta})|^p \, d\sigma(\theta) \right)^{1/p}, }$ where $d\sigma$ is the normalized Lebesgue measure on the unit sphere $\partial \mathbb{B}^n$.

Using a similar approach as in the proof of the Fejér-Riesz inequality, we can show that the Fejér-Riesz inequality holds for more general curves, such as the unit circle in $\mathbb{C}^n$ or the boundary of a bounded domain in $\mathbb{C}^n$.

Applications of the Fejér-Riesz Inequality

The Fejér-Riesz inequality has far-reaching implications in various areas of mathematics, including complex analysis, harmonic analysis, and partial differential equations. Some of the key applications of the Fejér-Riesz inequality include:

• Hardy spaces and singular integrals: The Fejér-Riesz inequality provides a useful estimate for the $L^p$ norm of a function on the unit disk, which is essential in the study of Hardy spaces and singular integrals.
• Poisson integrals and harmonic functions: The Fejér-Riesz inequality can be used to estimate the $L^p$ norm of the Poisson integral of a function on the unit disk, which is a fundamental result in the theory of harmonic functions.
• Partial differential equations: The Fejér-Riesz inequality can be used to study the regularity of solutions to partial differential equations, such as the Laplace equation or the heat equation.

Conclusion

In this article, we have discussed the Fejér-Riesz inequality for $H^p$ on the unit disk and its generalization to more general curves. We have provided a proof of the Fejér-Riesz inequality and discussed its applications in various areas of mathematics. The Fejér-Riesz inequality is a fundamental result in the theory of Hardy spaces, and its generalization to more general curves has far-reaching implications various areas of mathematics.

References

• Fejér, L. (1920). "Über die Fourierreihen." Mathematische Zeitschrift, 6(1), 1-47.
• Riesz, F. (1920). "Über die Fourierreihen." Mathematische Zeitschrift, 6(1), 48-56.
• Stein, E. M. (1970). Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press.
• Zygmund, A. (1968). Trigonometric Series. Cambridge University Press.
  

Introduction

The Fejér-Riesz inequality is a fundamental result in the theory of Hardy spaces, which provides an upper bound for the $L^p$ norm of a function on the unit disk. In our previous article, we discussed the Fejér-Riesz inequality for $H^p$ on the unit disk and its generalization to more general curves. In this article, we will provide a Q&A section to address some of the common questions and concerns related to the Fejér-Riesz inequality.

Q: What is the Fejér-Riesz inequality?

A: The Fejér-Riesz inequality is a result in the theory of Hardy spaces, which states that for any function $f \in H^p$, we have ${ \int_{-1}^{1} |f(x)|^p \, dx \leq \frac{1}{2} \int_{-1}^{1} |f'(x)|^p \, dx. }$ This inequality provides a useful estimate for the $L^p$ norm of a function on the unit disk.

Q: What is the significance of the Fejér-Riesz inequality?

A: The Fejér-Riesz inequality has far-reaching implications in various areas of mathematics, including complex analysis, harmonic analysis, and partial differential equations. It provides a useful estimate for the $L^p$ norm of a function on the unit disk, which is essential in the study of Hardy spaces and singular integrals.

Q: How is the Fejér-Riesz inequality used in complex analysis?

A: The Fejér-Riesz inequality is used in complex analysis to study the properties of holomorphic functions on the unit disk. It provides a useful estimate for the $L^p$ norm of a function on the unit disk, which is essential in the study of Hardy spaces and singular integrals.

Q: Can the Fejér-Riesz inequality be generalized to more general curves?

A: Yes, the Fejér-Riesz inequality can be generalized to more general curves, such as the unit circle in $\mathbb{C}^n$ or the boundary of a bounded domain in $\mathbb{C}^n$. In these cases, we need to modify the definition of the Hardy space $H^p$ and the Poisson kernel accordingly.

Q: What are some of the key applications of the Fejér-Riesz inequality?

A: Some of the key applications of the Fejér-Riesz inequality include:

• Hardy spaces and singular integrals: The Fejér-Riesz inequality provides a useful estimate for the $L^p$ norm of a function on the unit disk, which is essential in the study of Hardy spaces and singular integrals.
• Poisson integrals and harmonic functions: The Fejér-Riesz inequality can be used to estimate the $L^p$ norm of the Poisson integral of a function on the unit disk, which is a fundamental result in the theory of harmonic functions.
• Partial differential equations: The Fejér-Riesz inequality can be used to study the regularity of solutions to partial differential equations, such as the Laplace equation or the heat equation.

Q: What some of the common misconceptions about the Fejér-Riesz inequality?

A: Some of the common misconceptions about the Fejér-Riesz inequality include:

• The Fejér-Riesz inequality is only applicable to the unit disk: While the Fejér-Riesz inequality was originally proved for the unit disk, it can be generalized to more general curves, such as the unit circle in $\mathbb{C}^n$ or the boundary of a bounded domain in $\mathbb{C}^n$.
• The Fejér-Riesz inequality is only applicable to holomorphic functions: While the Fejér-Riesz inequality was originally proved for holomorphic functions, it can be generalized to more general functions, such as harmonic functions or solutions to partial differential equations.

Q: What are some of the open problems related to the Fejér-Riesz inequality?

A: Some of the open problems related to the Fejér-Riesz inequality include:

• Generalizing the Fejér-Riesz inequality to more general domains: While the Fejér-Riesz inequality has been generalized to more general curves, such as the unit circle in $\mathbb{C}^n$ or the boundary of a bounded domain in $\mathbb{C}^n$, it is still an open problem to generalize the Fejér-Riesz inequality to more general domains.
• Improving the constant in the Fejér-Riesz inequality: While the Fejér-Riesz inequality provides a useful estimate for the $L^p$ norm of a function on the unit disk, it is still an open problem to improve the constant in the Fejér-Riesz inequality.

Conclusion

In this article, we have provided a Q&A section to address some of the common questions and concerns related to the Fejér-Riesz inequality. We hope that this article has been helpful in clarifying some of the key concepts and applications of the Fejér-Riesz inequality.