Sharp estimates for the p-adic Hardy type operators on higher-dimensional product spaces.

Area of Science:

• * Number theory
• * Functional analysis
• * Harmonic analysis

Background:

• * The Hardy type operator is a fundamental tool in analysis.
• * p-adic analysis extends classical analysis to p-adic fields, offering new insights.
• * Lebesgue product spaces are crucial for studying function spaces.

Purpose of the Study:

• * To introduce and analyze the p-adic Hardy type operator.
• * To establish sharp bounds for this operator on p-adic Lebesgue product spaces.
• * To investigate weighted versions and related operators.

Main Methods:

• * Utilizing techniques from p-adic analysis and functional analysis.
• * Applying methods for determining sharp bounds of integral operators.
• * Characterizing boundedness conditions for weighted operators.

Main Results:

• * The sharp bound of the p-adic Hardy type operator on p-adic Lebesgue product spaces is obtained.
• * An analogous result is derived for p-adic Lebesgue product spaces with power weights.
• * A necessary and sufficient condition for the boundedness of the weighted p-adic Hardy type operator is established.

Conclusions:

• * The research extends the study of Hardy type operators to the p-adic setting.
• * The findings provide a comprehensive understanding of the operator's behavior on p-adic function spaces.
• * The work contributes to the development of p-adic harmonic analysis.