<p>In this paper, we introduce new spaces <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathcal A^p_{a,b}(\mathbb B_n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="script">A</mi> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> <mi>p</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">B</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of holomorphic functions on the unit ball <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathbb {B}_{n}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">B</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\mathbb {C}^{n}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> generalizing the classical Bergman spaces. We start by giving some properties of these spaces and the determination of the reproducing kernel in the case <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(p=2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, then we study the boundedness of the Bergman projections. Using the Berezin transform, a type of Bergman-Poincaré metric with an application on the space of bounded mean oscillation functions is given at the end of the paper.</p>